You would need to pull at least 13 cards from the deck to guarantee that you have at least four cards in your hand with the exact same suit.
In a standard deck of 52 playing cards, there are four suits: hearts, diamonds, clubs, and spades. To determine how many cards you would need to pull from the deck to ensure that you have at least four cards of the same suit in your hand, we can use the pigeonhole principle.
The worst-case scenario would be if you first draw three cards from each of the four suits, totaling 12 cards. In this case, you would have one card from each suit but not yet four cards of the same suit.
To ensure that you have at least four cards of the same suit, you would need to draw one additional card. By the pigeonhole principle, this card will necessarily match one of the suits already present in your hand, completing a set of four cards of the same suit.
Therefore, you would need to pull at least 13 cards from the deck to guarantee that you have at least four cards in your hand with the exact same suit.
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Solve the following equation.
37+w=5 w-27
The value of the equation is 16.
To solve the equation 37 + w = 5w - 27, we'll start by isolating the variable w on one side of the equation. Let's go step by step:
We begin with the equation 37 + w = 5w - 27.
First, let's get rid of the parentheses by removing them.
37 + w = 5w - 27
Next, we can simplify the equation by combining like terms.
w - 5w = -27 - 37
-4w = -64
Now, we want to isolate the variable w. To do so, we divide both sides of the equation by -4.
(-4w)/(-4) = (-64)/(-4)
w = 16
After simplifying and solving the equation, we find that the value of w is 16.
To check our solution, we substitute w = 16 back into the original equation:
37 + w = 5w - 27
37 + 16 = 5(16) - 27
53 = 80 - 27
53 = 53
The equation holds true, confirming that our solution of w = 16 is correct.
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How can you clear the equation x/3 + 1 = 1/6 of fractions? a. Multiply each term by 3 b. Divide each term by 6 c. Divide each term by 3 d. Multiply each term by 6 e. Subtract 1 from each side.
we can solve for x by dividing both sides by 2:x = -5/2 Therefore, the answer is to multiply each term by 6 to clear the equation of fractions.
To clear the equation x/3 + 1 = 1/6 of fractions, you have to multiply each term by 6.
This will eliminate the fractions and make it easier to solve the equation.
To solve the equation x/3 + 1 = 1/6, we need to get rid of the fractions.
One way to do this is to multiply each term by the least common multiple (LCM) of the denominators, which in this case is 6.
By doing this, we can clear the equation of fractions and make it easier to solve.
First, we multiply each term by 6 to eliminate the fractions: x/3 + 1 = 1/6
becomes 6(x/3) + 6(1) = 6(1/6)
Simplifying this equation, we get:
2x + 6 = 1
Now we can isolate the variable by subtracting 6 from both sides:
2x + 6 - 6 = 1 - 6
Simplifying further, we get:
2x = -5
Finally, we can solve for x by dividing both sides by 2:x = -5/2Therefore, the answer is to multiply each term by 6 to clear the equation of fractions.
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Nine subtracted from nine times a number is - 108 . What is the number? A) Translate the statement above into an equation that you can solve to answer this question. Do not solve it yet. Use x as your variable. The equation is B) Solve your equation in part [A] for x.
The equation for the given problem is 9x - 9 = -108. To solve for x, we need to simplify the equation and isolate the variable.
Let's break down the problem step by step.
The first part states "nine times a number," which can be represented as 9x, where x is the unknown number.
The next part says "nine subtracted from," so we subtract 9 from 9x, resulting in 9x - 9.
Finally, the problem states that this expression is equal to -108, giving us the equation 9x - 9 = -108.
To solve for x, we need to isolate the variable on one side of the equation. We can do this by performing inverse operations.
First, we add 9 to both sides of the equation to eliminate the -9 on the left side, resulting in 9x = -99.
Next, we divide both sides by 9 to isolate x. By dividing -99 by 9, we find that x = -11.
Therefore, the number we're looking for is -11.
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Find the area of region bounded by f(x)=8−7x 2
,g(x)=x, from x=0 and x−1. Show all work, doing, all integration by hand. Give your final answer in friction form (not a decimal),
The area of the region bounded by the curves is 15/2 - 7/3, which is a fractional form. To find the area of the region bounded by the curves f(x) = 8 - 7x^2 and g(x) = x from x = 0 to x = 1, we can calculate the definite integral of the difference between the two functions over the interval [0, 1].
First, let's set up the integral for the area:
Area = ∫[0 to 1] (f(x) - g(x)) dx
= ∫[0 to 1] ((8 - 7x^2) - x) dx
Now, we can simplify the integrand:
Area = ∫[0 to 1] (8 - 7x^2 - x) dx
= ∫[0 to 1] (8 - 7x^2 - x) dx
= ∫[0 to 1] (8 - 7x^2 - x) dx
Integrating term by term, we have:
Area = [8x - (7/3)x^3 - (1/2)x^2] evaluated from 0 to 1
= [8(1) - (7/3)(1)^3 - (1/2)(1)^2] - [8(0) - (7/3)(0)^3 - (1/2)(0)^2]
= 8 - (7/3) - (1/2)
Simplifying the expression, we get:
Area = 8 - (7/3) - (1/2) = 15/2 - 7/3
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Writing Equations Parallel & Perpendicular Lines.
1. Write the slope-intercept form of the equation of the line described. Through: (2,2), parallel y= x+4
2. Through: (4,3), Parallel to x=0.
3.Through: (1,-5), Perpendicular to Y=1/8x + 2
Equation of the line described: y = x + 4
Slope of given line y = x + 4 is 1
Therefore, slope of parallel line is also 1
Using the point-slope form of the equation of a line,
we have y - y1 = m(x - x1),
where (x1, y1) = (2, 2)
Substituting the values, we get
y - 2 = 1(x - 2)
Simplifying the equation, we get
y = x - 1
Therefore, slope-intercept form of the equation of the line is
y = x - 12.
Equation of the line described:
x = 0
Since line is parallel to the y-axis, slope of the line is undefined
Therefore, the equation of the line is x = 4.3.
Equation of the line described:
y = (1/8)x + 2
Slope of given line y = (1/8)x + 2 is 1/8
Therefore, slope of perpendicular line is -8
Using the point-slope form of the equation of a line,
we have y - y1 = m(x - x1),
where (x1, y1) = (1, -5)
Substituting the values, we get
y - (-5) = -8(x - 1)
Simplifying the equation, we get y = -8x - 3
Therefore, slope-intercept form of the equation of the line is y = -8x - 3.
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in tests of significance about an unknown parameter, what does the test statistic represent? group of answer choices a measure of compatibility between the null hypothesis and the data. a measure of compatibility between the null and alternative hypotheses. the value of the unknown parameter under the alternative hypothesis. the value of the unknown parameter under the null hypothesis.
The test statistic represents a measure of compatibility between the null hypothesis and the data in tests of significance about an unknown parameter.
In hypothesis testing, we compare the observed data to what we would expect if the null hypothesis were true. The test statistic is a calculated value that quantifies the extent to which the observed data deviates from what is expected under the null hypothesis.
It is important to note that the test statistic is not directly related to the value of the unknown parameter. Instead, it provides a measure of how well the data align with the null hypothesis.
By comparing the test statistic to critical values or p-values, we can determine the level of evidence against the null hypothesis. If the test statistic falls in the critical region or the p-value is below the chosen significance level, we reject the null hypothesis in favor of the alternative hypothesis.
Therefore, the test statistic serves as a measure of compatibility between the null hypothesis and the data, helping us assess the strength of evidence against the null hypothesis.
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Select the correct answer from each drop-down menu. a teacher created two-way tables for four different classrooms. the tables track whether each student was a boy or girl and whether they were in art class only, music class only, both classes, or neither class. classroom 1 art only music only both neither boys 2 4 5 2 girls 5 4 7 1 classroom 2 art only music only both neither boys 4 1 3 4 girls 1 4 5 2 classroom 3 art only music only both neither boys 3 4 1 3 girls 2 3 4 0 classroom 4 art only music only both neither boys 4 5 3 2 girls 6 3 4 3 classroom has an equal number of boys and girls. classroom has the smallest number of students in music class. classroom has the largest number of students who are not in art class or music class. classroom has the largest number of students in art class but not music class.
Classroom 2 has an equal number of boys and girls.Classroom 2 has the smallest number of students in music class.Classroom 1 has the largest number of students who are not in art class or music class.Classroom 1 has the largest number of students in art class but not music class.
To find which class has an equal number of boys and girls, we can examine each class. The total number of boys and girls are:
Classroom 1: 13 boys, 17 girls
Classroom 2: 12 boys, 12 girls
Classroom 3: 11 boys, 9 girls
Classroom 4: 14 boys, 16 girls
Classrooms 1 and 2 do not have an equal number of boys and girls.
Classroom 4 has more girls than boys and Classroom 3 has more boys than girls.
Therefore, Classroom 2 is the only class that has an equal number of boys and girls.
We can find the smallest number of students in music class by finding the smallest total in the "music only" column. Classroom 2 has the smallest total in this column with 8 students. Therefore, Classroom 2 has the smallest number of students in music class.We can find which classroom has the largest number of students who are not in art class or music class by finding the largest total in the "neither" column.
Classroom 1 has the largest total in this column with 3 students. Therefore, Classroom 1 has the largest number of students who are not in art class or music class.We can find which classroom has the largest number of students in art class but not music class by finding the largest total in the "art only" column and subtracting the "both" column from it. Classroom 1 has the largest total in the "art only" column with 7 students and also has 5 students in the "both" column.
Therefore, 7 - 5 = 2 students are in art class but not music class in Classroom 1.
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Evaluate the derivative of the function f(t)=7t+4/5t−1 at the point (3,25/14 )
The derivative of the function f(t) = (7t + 4)/(5t − 1) at the point (3, 25/14) is -3/14.At the point (3, 25/14), the function f(t) = (7t + 4)/(5t − 1) has a derivative of -3/14, indicating a negative slope.
To evaluate the derivative of the function f(t) = (7t + 4) / (5t - 1) at the point (3, 25/14), we'll first find the derivative of f(t) and then substitute t = 3 into the derivative.
To find the derivative, we can use the quotient rule. Let's denote f'(t) as the derivative of f(t):
f(t) = (7t + 4) / (5t - 1)
f'(t) = [(5t - 1)(7) - (7t + 4)(5)] / (5t - 1)^2
Simplifying the numerator:
f'(t) = (35t - 7 - 35t - 20) / (5t - 1)^2
f'(t) = (-27) / (5t - 1)^2
Now, substitute t = 3 into the derivative:
f'(3) = (-27) / (5(3) - 1)^2
= (-27) / (15 - 1)^2
= (-27) / (14)^2
= (-27) / 196
So, the derivative of f(t) at the point (3, 25/14) is -27/196.The derivative represents the slope of the tangent line to the curve of the function at a specific point.
In this case, the slope of the function f(t) = (7t + 4) / (5t - 1) at t = 3 is -27/196, indicating a negative slope. This suggests that the function is decreasing at that point.
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Science
10 Consider the following statement.
A student measured the pulse rates
(beats per minute) of five classmates
before and after running. Before they
ran, the average rate was 70 beats
per minute, and after they ran,
the average was 150 beats per minute.
The underlined portion of this statement
is best described as
Ja prediction.
Ka hypothesis.
L an assumption.
M an observation.
It is an observation rather than a prediction, hypothesis, or assumption.
The underlined portion of the statement, "Before they ran, the average rate was 70 beats per minute, and after they ran, the average was 150 beats per minute," is best described as an observation.
An observation is a factual statement made based on the direct gathering of data or information. In this case, the student measured the pulse rates of five classmates before and after running, and the statement reports the average rates observed before and after the activity.
It does not propose a cause-and-effect relationship or make any assumptions or predictions. Instead, it presents the actual measured values and provides information about the observed change in pulse rates. Therefore, it is an observation rather than a prediction, hypothesis, or assumption.
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Question
A student measured the pulse rates
(beats per minute) of five classmates
before and after running. Before they
ran, the average rate was 70 beats
per minute, and after they ran,
the average was 150 beats per minute.
The underlined portion of this statement
is best described as
Ja prediction.
Ka hypothesis.
L an assumption.
M an observation.
Assume a random variable Z has a standard normal distribution (mean 0 and standard deviation 1). Answer the questions below by referring to the standard normal distribution table provided in the formula sheet. a) The probability that Z lies between -1.05 and 1.76 is [ Select ] to 4 decimal places. b) The probability that Z is less than -1.05 or greater than 1.76 is [ Select ] to 4 decimal places. c) What is the value of Z if only 1.7% of all possible Z values are larger than it? [ Select ] keep to 2 decimal places.
a) The probability that Z lies between -1.05 and 1.76 is 0.8664 to 4 decimal places.
b) The probability that Z is less than -1.05 or greater than 1.76 is 0.1588 to 4 decimal places.
c) The value of Z, where only 1.7% of all possible Z values are larger than it, is 1.41 to 2 decimal places.
a) To find the probability that Z lies between -1.05 and 1.76, we need to find the area under the standard normal distribution curve between these two values. By using the standard normal distribution table, we can find the corresponding probabilities for each value and subtract them. The probability is calculated as 0.8664.
b) The probability that Z is less than -1.05 or greater than 1.76 can be found by calculating the sum of the probabilities of Z being less than -1.05 and Z being greater than 1.76. Using the standard normal distribution table, we find the probabilities for each value and add them together. The probability is calculated as 0.1588.
c) If only 1.7% of all possible Z values are larger than a certain Z value, we need to find the Z value corresponding to the 98.3rd percentile (100% - 1.7%). Using the standard normal distribution table, we can look up the value closest to 98.3% and find the corresponding Z value. The Z value is calculated as 1.41.
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show all the work please!
105. Find the given distances between points \( P, Q, R \), and \( S \) on a number line, with coordinates \( -4,-1,8 \), and 12 , respectively. \[ d(P, Q) \]
The distance between points P and Q on the number line can be found by taking the absolute value of the difference of their coordinates. In this case, the distance between P and Q is 3.
To find the distance between points P and Q on the number line, we can take the absolute value of the difference of their coordinates. The coordinates of point P is -4, and the coordinates of point Q is -1.
Using the formula for distance between two points on the number line, we have:
d(P, Q) = |(-1) - (-4)|
Simplifying the expression inside the absolute value:
d(P, Q) = |(-1) + 4|
Calculating the sum inside the absolute value:
d(P, Q) = |3|
Taking the absolute value of 3:
d(P, Q) = 3
Therefore, the distance between points P and Q on the number line is 3.
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Question 5 (20 points ) (a) in a sample of 12 men the quantity of hemoglobin in the blood stream had a mean of 15 / and a standard deviation of 3 g/ dlfind the 99% confidence interval for the population mean blood hemoglobin . (round your final answers to the nearest hundredth ) the 99% confidence interval is. dot x pm t( s sqrt n )15 pm1
The 99% confidence interval for the population mean blood hemoglobin is 12.31 < μ < 17. 69.
Given that,
Hemoglobin concentration in a sample of 12 men had a mean of 15 g/dl and a standard deviation of 3 g/dl.
We have to find the 99% confidence interval for the population mean blood hemoglobin.
We know that,
Let n = 12
Mean X = 15 g/dl
Standard deviation s = 3 g/dl
The critical value α = 0.01
Degree of freedom (df) = n - 1 = 12 - 1 = 11
[tex]t_c[/tex] = [tex]z_{1-\frac{\alpha }{2}, n-1}[/tex] = 3.106
Then the formula of confidential interval is
= (X - [tex]t_c\times \frac{s}{\sqrt{n} }[/tex] , X + [tex]t_c\times \frac{s}{\sqrt{n} }[/tex] )
= (15- 3.106 × [tex]\frac{3}{\sqrt{12} }[/tex], 15 + 3.106 × [tex]\frac{3}{\sqrt{12} }[/tex] )
= (12.31, 17.69)
Therefore, The 99% confidence interval for the population mean blood hemoglobin is 12.31 < μ < 17. 69.
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Set Identities:
Show that the following are true:(show work)
1. A−B = A−(A∩B)
2. A∩B = A∪B
3. (A−B)−C = (A−C)−(B−C)
NOTE : remember that to show two sets are equal, we must show
th
To show that A−B = A−(A∩B), we need to show that A−B is a subset of A−(A∩B) and that A−(A∩B) is a subset of A−B. Let x be an element of A−B. This means that x is in A and x is not in B.
By definition of set difference, if x is not in B, then x is not in A∩B. So, x is in A−(A∩B), which shows that A−B is a subset of A−(A∩B). Let x be an element of A−(A∩B). This means that x is in A and x is not in A∩B. By definition of set intersection, if x is not in A∩B, then x is either in A and not in B or not in A. So, x is in A−B, which shows that A−(A∩B) is a subset of A−B. Therefore, we have shown that A−B = A−(A∩B).
2. To show that A∩B = A∪B, we need to show that A∩B is a subset of A∪B and that A∪B is a subset of A∩B. Let x be an element of A∩B. This means that x is in both A and B, so x is in A∪B. Therefore, A∩B is a subset of A∪B. Let x be an element of A∪B. This means that x is in A or x is in B (or both). If x is in A, then x is also in A∩B, and if x is in B, then x is also in A∩B. Therefore, A∪B is a subset of A∩B. Therefore, we have shown that A∩B = A∪B.
3. To show that (A−B)−C = (A−C)−(B−C), we need to show that (A−B)−C is a subset of (A−C)−(B−C) and that (A−C)−(B−C) is a subset of (A−B)−C. Let x be an element of (A−B)−C. This means that x is in A but not in B, and x is not in C. By definition of set difference, if x is not in C, then x is in A−C. Also, if x is in A but not in B, then x is either in A−C or in B−C. However, x is not in B−C, so x is in A−C.
Therefore, x is in (A−C)−(B−C), which shows that (A−B)−C is a subset of (A−C)−(B−C). Let x be an element of (A−C)−(B−C). This means that x is in A but not in C, and x is not in B but may or may not be in C. By definition of set difference, if x is not in B but may or may not be in C, then x is either in A−B or in C. However, x is not in C, so x is in A−B. Therefore, x is in (A−B)−C, which shows that (A−C)−(B−C) is a subset of (A−B)−C. Therefore, we have shown that (A−B)−C = (A−C)−(B−C).
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The lengths of the legs of a right triangle are given below. Find the length of the hypotenuse. a=55,b=132 The length of the hypotenuse is units.
The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem. In this case, with the lengths of the legs being a = 55 and b = 132, the length of the hypotenuse is calculated as c = √(a^2 + b^2). Therefore, the length of the hypotenuse is approximately 143.12 units.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, it can be expressed as c^2 = a^2 + b^2.
In this case, the lengths of the legs are given as a = 55 and b = 132. Plugging these values into the formula, we have c^2 = 55^2 + 132^2. Evaluating this expression, we find c^2 = 3025 + 17424 = 20449.
To find the length of the hypotenuse, we take the square root of both sides of the equation, yielding c = √20449 ≈ 143.12. Therefore, the length of the hypotenuse is approximately 143.12 units.
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Use transformations of the graph of f(x)=e^x to graph the given function. Be sure to the give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm the hand-drawn graphs. g(x)=e^(x−5). Determine the transformations that are needed to go from f(x)=e^x to the given graph. Select all that apply. A. shrink vertically B. shift 5 units to the left C. shift 5 units downward D. shift 5 units upward E. reflect about the y-axis F. reflect about the x-axis G. shrink horizontally H. stretch horizontally I. stretch vertically
Use transformations of the graph of f(x)=e^x to graph the given function. Be sure to the give equations of the asymptotes. Thus, option C, A, H and I are the correct answers.
The given function is g(x) = e^(x - 5). To graph the function, we need to determine the transformations that are needed to go from f(x) = e^x to g(x) = e^(x - 5).
Transformations are described below:Since the x-axis value is increased by 5, the graph must shift 5 units to the right. Therefore, option B is incorrect. The graph shifts downwards by 5 units since the y-axis value of the graph is reduced by 5 units.
Therefore, the correct option is C.
The graph gets shrunk vertically since it becomes narrower. Therefore, option A is correct.Since there are no y-axis changes, the graph is not reflected about the y-axis. Therefore, the correct option is not E.Since there are no x-axis changes, the graph is not reflected about the x-axis. Therefore, the correct option is not F.
There is no horizontal compression because the horizontal distance between the points remains the same. Therefore, the correct option is not G.There is a horizontal expansion since the graph is stretched out. Therefore, the correct option is H.
There is a vertical expansion since the graph is stretched out. Therefore, the correct option is I.Using the transformations, the new graph will be as shown below:Asymptotes:
There are no horizontal asymptotes for the function. Range: (0, ∞)Domain: (-∞, ∞)The graph shows that the function is an increasing function. Therefore, the range of the function is (0, ∞) and the domain is (-∞, ∞). Thus, option C, A, H and I are the correct answers.
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Solve the following linear system of equations by using: A) Gaussian elimination: B) Gaussian Jordan elimination: C) Doolittle LU decomposition: D) Croute LU decomposition: E) Chelosky LU decomposition: x−2y+3z=4
2x+y−4z=3
−3x+4y−z=−2
By Gaussian elimination, the solution for a given system of linear equations is (x, y, z) = (2/15, 17/15, 5/3).
Given the linear system of equations:
x − 2y + 3z = 4 ... (i)
2x + y − 4z = 3 ... (ii)
− 3x + 4y − z = − 2 ... (iii)
Gaussian elimination:
In Gaussian elimination, the given system of equations is transformed into an equivalent upper triangular system of equations by performing elementary row operations. The steps to solve the given system of equations by Gaussian elimination are as follows:
Step 1: Write the augmented matrix of the given system of equations.
[tex][A|B] = \[\left[\begin{matrix}1 & -2 & 3 \\2 & 1 & -4 \\ -3 & 4 & -1\end{matrix}\middle| \begin{matrix} 4 \\ 3 \\ -2 \end{matrix}\right]\][/tex]
Step 2: Multiply R1 by 2 and subtract from R2, and then multiply R1 by -3 and add to R3. The resulting matrix is:
[tex]\[\left[\begin{matrix}1 & -2 & 3 \\0 & 5 & -10 \\ 0 & -2 & 8\end{matrix}\middle| \begin{matrix} 4 \\ 5 \\ -10 \end{matrix}\right]\][/tex]
Step 3: Multiply R2 by 2 and add to R3. The resulting matrix is:
[tex]\[\left[\begin{matrix}1 & -2 & 3 \\0 & 5 & -10 \\ 0 & 0 & -12\end{matrix}\middle| \begin{matrix} 4 \\ 5 \\ -20 \end{matrix}\right]\][/tex]
Step 4: Solve for z, y, and x respectively from the resulting matrix. The solution is:
z = 20/12 = 5/3y = (5 + 2z)/5 = 17/15x = (4 - 3z + 2y)/1 = 2/15
Therefore, the solution to the given system of equations by Gaussian elimination is:(x, y, z) = (2/15, 17/15, 5/3)
Gaussian elimination is a useful method of solving a system of linear equations. It involves performing elementary row operations on the augmented matrix of the system to obtain a triangular form. The unknown variables can then be solved for by back-substitution. In this problem, Gaussian elimination was used to solve the given system of linear equations. The solution is (x, y, z) = (2/15, 17/15, 5/3).
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can
somone help
Solve for all values of \( y \) in simplest form. \[ |y-12|=16 \]
The final solution is the union of all possible solutions. The solution of the given equation is [tex]\[y=28, -4\].[/tex]
Given the equation [tex]\[|y-12|=16\][/tex]
We need to solve for all values of y in the simplest form.
Given the equation [tex]\[|y-12|=16\][/tex]
We know that,If [tex]\[a>0\][/tex]then, [tex]\[|x|=a\][/tex] means[tex]\[x=a\] or \[x=-a\][/tex]
If [tex]\[a<0\][/tex] then,[tex]\[|x|=a\][/tex] means no solution.
Now, for the given equation, [tex]|y-12|=16[/tex] is of the form [tex]\[|x-a|=b\][/tex] where a=12 and b=16
Therefore, y-12=16 or y-12=-16
Now, solving for y,
y-12=16
y=16+12
y=28
y-12=-16
y=-16+12
y=-4
Therefore, the solution of the given equation is y=28, -4
We can solve the given equation |y-12|=16 by using the concept of modulus function. We write the modulus function in terms of positive or negative sign and solve the equation by taking two cases, one for positive and zero values of (y - 12), and the other for negative values of (y - 12). The final solution is the union of all possible solutions. The solution of the given equation is y=28, -4.
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A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x,y,z)=35−3(x 2
+y 2
+z 2
) ∘
C. Use the fact that heat flow is given by the vector field F=−K∇w and the rate of heat flow across a surface S within the solid is given by −K∬ S
∇wdS. Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper (K=400 kW/(m⋅K)) (Use symbolic notation and fractions where needed.) −K∬ S
∇wdS= kW
The rate of heat flow out of the sphere is 0 kW.
To find the rate of heat flow out of a sphere of radius 1 inside a large cube of copper, we need to calculate the surface integral of the gradient of the temperature function w(x, y, z) over the surface of the sphere.
First, let's calculate the gradient of w(x, y, z):
∇w = (∂w/∂x)i + (∂w/∂y)j + (∂w/∂z)k
∂w/∂x = -6x
∂w/∂y = -6y
∂w/∂z = -6z
So, ∇w = -6xi - 6yj - 6zk
The surface integral of ∇w over the surface of the sphere can be calculated using spherical coordinates. In spherical coordinates, the surface element dS is given by dS = r^2sinθdθdφ, where r is the radius of the sphere (1 in this case), θ is the polar angle, and φ is the azimuthal angle.
Since the surface is a sphere of radius 1, the limits of integration for θ are 0 to π, and the limits for φ are 0 to 2π.
Now, let's calculate the surface integral:
−K∬ S ∇wdS = −K∫∫∫ ρ^2sinθdθdφ
−K∬ S ∇wdS = −K∫₀²π∫₀ᴨ√(ρ²sin²θ)ρdθdφ
−K∬ S ∇wdS = −K∫₀²π∫₀ᴨρ²sinθdθdφ
−K∬ S ∇wdS = −K∫₀²π∫₀ᴨρ²sinθ(-6ρsinθ)dθdφ
−K∬ S ∇wdS = 6K∫₀²π∫₀ᴨρ³sin²θdθdφ
Since we are integrating over the entire sphere, the limits for ρ are 0 to 1.
−K∬ S ∇wdS = 6K∫₀²π∫₀ᴨρ³sin²θdθdφ
−K∬ S ∇wdS = 6K∫₀²π∫₀ᴨ(ρ³/2)(1 - cos(2θ))dθdφ
−K∬ S ∇wdS = 6K∫₀²π[(ρ³/2)(θ - (1/2)sin(2θ))]|₀ᴨdφ
−K∬ S ∇wdS = 6K∫₀²π[(1/2)(θ - (1/2)sin(2θ))]|₀ᴨdφ
−K∬ S ∇wdS = 6K∫₀²π[(1/2)(0 - (1/2)sin(2(0)))]dφ
−K∬ S ∇wdS = 6K∫₀²π(0)dφ
−K∬ S ∇wdS = 0
Therefore, the rate of heat flow out of the sphere is 0 kW.
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suppose you sampled 14 working students and obtained the following data representing, number of hours worked per week {35, 20, 20, 60, 20, 13, 12, 35, 25, 15, 20, 35, 20, 15}. how many students would be in the 3rd class if the width is 15 and the first class ends at 15 hours per week? select one: 6 5 3 4
To determine the number of students in the third class, we need to first calculate the boundaries of each class interval based on the given width and starting point.
Given that the first class ends at 15 hours per week, we can construct the class intervals as follows:
Class 1: 0 - 15
Class 2: 16 - 30
Class 3: 31 - 45
Class 4: 46 - 60
Now we can examine the data and count how many values fall into each class interval:
Class 1: 13, 12, 15 --> 3 students
Class 2: 20, 20, 20, 25, 15, 20, 15 --> 7 students
Class 3: 35, 35, 35, 60, 35 --> 5 students
Class 4: 20 --> 1 student
Therefore, there are 5 students in the third class.
In summary, based on the given data and the class intervals with a width of 15 starting at 0-15, there are 5 students in the third class.
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If 30 locusts eat 429 grams of grass in a week. how many days will take 21 locusts to consume 429grams of grass if they eat at the same rate
The given statement is that 30 locusts consume 429 grams of grass in a week.It would take 10 days for 21 locusts to eat 429 grams of grass if they eat at the same rate as 30 locusts.
A direct proportionality exists between the number of locusts and the amount of grass they consume. Let "a" be the time required for 21 locusts to eat 429 grams of grass. Then according to the statement given, the time required for 30 locusts to eat 429 grams of grass is 7 days.
Let's first find the amount of grass consumed by 21 locusts in 7 days:Since the number of locusts is proportional to the amount of grass consumed, it can be expressed as:
21/30 = 7/a21
a = 30 × 7
a = 30 × 7/21
a = 10
Therefore, it would take 10 days for 21 locusts to eat 429 grams of grass if they eat at the same rate as 30 locusts.
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9) Find the inverse of the function. f(x)=3x+2 f −1
(x)= 3
1
x− 3
2
f −1
(x)=5x+6
f −1
(x)=−3x−2
f −1
(x)=2x−3
10) Find the solution to the system of equations. (4,−2)
(−4,2)
(2,−4)
(−2,4)
11) Which is the standard form equation of the ellipse? 8x 2
+5y 2
−32x−20y=28 10
(x−2) 2
+ 16
(y−2) 2
=1 10
(x+2) 2
+ 16
(y+2) 2
=1
16
(x−2) 2
+ 10
(y−2) 2
=1
16
(x+2) 2
+ 10
(y+2) 2
=1
9) Finding the inverse of a function is quite simple, and it involves swapping the input with the output in the function equation. Here's how the process is carried out;f(x)=3x+2Replace f(x) with y y=3x+2 Swap x and y x=3y+2 Isolate y 3y=x−2 Divide by 3 y=x−23 Solve for y y=13(x−3)Therefore f −1(x)= 3
1
x− 3
2
The inverse of a function is a new function that maps the output of the original function to its input. The inverse function is a reflection of the original function across the line y = x.
The graph of a function and its inverse are reflections of each other over the line y = x. To find the inverse of a function, swap the x and y variables, then solve for y in terms of x.10) The system of equations given is(4, −2)(−4, 2)We have to find the solution to the given system of equations. The solution to a system of two equations in two variables is an ordered pair (x, y) that satisfies both equations.
One of the methods of solving a system of equations is to plot the equations on a graph and find the point of intersection of the two lines. This is where both lines cross each other. The intersection point is the solution of the system of equations. From the given system of equations, it is clear that the two equations represent perpendicular lines. This is because the product of their slopes is -1.
The lines have opposite slopes which are reciprocals of each other. Thus, the only solution to the given system of equations is (4, −2).11) The equation of an ellipse is generally given as;((x - h)2/a2) + ((y - k)2/b2) = 1The ellipse has its center at (h, k), and the major axis lies along the x-axis, and the minor axis lies along the y-axis.
The standard form equation of an ellipse is given as;(x2/a2) + (y2/b2) = 1where a and b are the length of major and minor axis respectively.8x2 + 5y2 − 32x − 20y = 28This equation can be rewritten as;8(x2 - 4x) + 5(y2 - 4y) = -4Now we complete the square in x and y to get the equation in standard form.8(x2 - 4x + 4) + 5(y2 - 4y + 4) = -4 + 32 + 20This can be simplified as follows;8(x - 2)2 + 5(y - 2)2 = 48Divide by 48 on both sides, we have;(x - 2)2/6 + (y - 2)2/9.6 = 1Thus, the standard form equation of the ellipse is 16(x - 2)2 + 10(y - 2)2 = 96.
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Expand each binomial.
(3 y-11)⁴
Step-by-step explanation:
mathematics is a equation of mind.
Find all equilibria of y ′
=2y−3y 2
, and determine whether each is locally stable or unstable. Then sketch the phase plot and describe the long term behavior of the system. Find the eigenvectors and corresponding eigenvalues of the given matrices. (a) ( 1
2
2
1
) (b) ( 1
1
−1
1
) (c) ( −1
0
2
−1
)
We obtain the eigenvector: v2 = [x, y] = [(-42 + 24√37) / (5√37), (-3√37 + 8) / 5]. These are the eigenvectors corresponding to the eigenvalues of the matrix.
To find the equilibria of the system and determine their stability, we need to solve the equation y' = 2y - 3y^2 for y. Setting y' equal to zero gives us: 0 = 2y - 3y^2. Next, we factor out y: 0 = y(2 - 3y). Setting each factor equal to zero, we find two possible equilibria: y = 0 or 2 - 3y = 0. For the second equation, we solve for y: 2 - 3y = 0, y = 2/3. So the equilibria are y = 0 and y = 2/3. To determine the stability of each equilibrium, we can evaluate the derivative of y' with respect to y, which is the second derivative of the original equation: y'' = d/dy(2y - 3y^2 = 2 - 6y
Now we substitute the values of y for each equilibrium: For y = 0
y'' = 2 - 6(0)= 2. Since y'' is positive, the equilibrium at y = 0 is unstable.
For y = 2/3: y'' = 2 - 6(2/3)= 2 - 4= -2. Since y'' is negative, the equilibrium at y = 2/3 is locally stable. Now let's sketch the phase plot and describe the long-term behavior of the system: The phase plot is a graph that shows the behavior of the system over time. We plot y on the vertical axis and y' on the horizontal axis. We have two equilibria: y = 0 and y = 2/3.
For y < 0, y' is positive, indicating that the system is moving away from the equilibrium at y = 0. As y approaches 0, y' approaches 2, indicating that the system is moving upward. For 0 < y < 2/3, y' is negative, indicating that the system is moving towards the equilibrium at y = 2/3. As y approaches 2/3, y' approaches -2, indicating that the system is moving downward. For y > 2/3, y' is positive, indicating that the system is moving away from the equilibrium at y = 2/3. As y approaches infinity, y' approaches positive infinity, indicating that the system is moving upward.
Based on this analysis, the long-term behavior of the system can be described as follows: For initial conditions with y < 0, the system moves away from the equilibrium at y = 0 and approaches positive infinity. For initial conditions with 0 < y < 2/3, the system moves towards the equilibrium at y = 2/3 and settles at this stable equilibrium. For initial conditions with y > 2/3, the system moves away from the equilibrium at y = 2/3 and approaches positive infinity.
Now let's find the eigenvectors and corresponding eigenvalues for the given matrices:(a) Matrix:
| 1/2 2 |
| 2 1 |
To find the eigenvectors and eigenvalues, we solve the equation (A - λI)v = 0, where A is the matrix, λ is the eigenvalue, I is the identity matrix, and v is the eigenvector. Substituting the given matrix into the equation, we have:
| 1/2 - λ 2 | | x | | 0 |
| 2 1 - λ | | y | = | 0 |
Expanding and rearranging, we get the following system of equations:
(1/2 - λ)x + 2y = 0, 2x + (1 - λ)y = 0. Solving this system of equations, we find: (1/2 - λ)x + 2y = 0 [1], 2x + (1 - λ)y = 0 [2]. From equation [1], we can solve for x in terms of y: x = -2y / (1/2 - λ). Substituting this value of x into equation [2], we get: 2(-2y / (1/2 - λ)) + (1 - λ)y = 0. Simplifying further:
-4y / (1/2 - λ) + (1 - λ)y = 0
-4y + (1/2 - λ - λ/2 + λ^2)y = 0
(-7/2 - 3λ/2 + λ^2)y = 0
For this equation to hold, either y = 0 (giving a trivial solution) or the expression in the parentheses must be zero: -7/2 - 3λ/2 + λ^2 = 0. Rearranging the equation: λ^2 - 3λ/2 - 7/2 = 0. To find the eigenvalues, we can solve this quadratic equation. Using the quadratic formula: λ = (-(-3/2) ± √((-3/2)^2 - 4(1)(-7/2))) / (2(1)). Simplifying further:
λ = (3/2 ± √(9/4 + 28/4)) / 2
λ = (3 ± √37) / 4
So the eigenvalues for matrix (a) are λ = (3 + √37) / 4 and λ = (3 - √37) / 4.
To find the eigenvectors corresponding to each eigenvalue, we substitute the eigenvalues back into the system of equations: For λ = (3 + √37) / 4: (1/2 - (3 + √37) / 4)x + 2y = 0 [1], 2x + (1 - (3 + √37) / 4)y = 0 [2]
Simplifying equation [1]: (-1/2 - √37/4)x + 2y = 0
Simplifying equation [2]: 2x + (-3/4 - √37/4)y = 0
For λ = (3 - √37) / 4, the system of equations would be slightly different:
(-1/2 + √37/4)x + 2y = 0 [1]
2x + (-3/4 + √37/4)y = 0 [2]
Solving these systems of equations will give us the corresponding eigenvectors.
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Find dy/dx for the equation below. 8x 4 +6 squ. root of xy =8y 2
The derivative of the given equation with respect to x is (32x3 + 3√y) / (8y - 3xy(-1/2)).
The given equation is:8x4 + 6√xy = 8y2We are to find dy/dx.To solve this, we need to use implicit differentiation on both sides of the equation.
Using the chain rule, we have: (d/dx)(8x4) + (d/dx)(6√xy) = (d/dx)(8y2).
Simplifying the left-hand side by using the power rule and the chain rule, we get: 32x3 + 3√y + 6x(1/2) * y(-1/2) * (dy/dx) = 16y(dy/dx).
Simplifying the right-hand side, we get: (d/dx)(8y2) = 16y(dy/dx).
Simplifying both sides of the equation, we have:32x3 + 3√y + 3xy(-1/2) * (dy/dx) = 8y(dy/dx)32x3 + 3√y = (8y - 3xy(-1/2))(dy/dx)dy/dx = (32x3 + 3√y) / (8y - 3xy(-1/2))This is the main answer.
we can provide a brief explanation on the topic of implicit differentiation and provide a step-by-step solution. Implicit differentiation is a method used to find the derivative of a function that is not explicitly defined.
This is done by differentiating both sides of an equation with respect to x and then solving for the derivative. In this case, we used implicit differentiation to find dy/dx for the given equation.
We used the power rule and the chain rule to differentiate both sides and then simplified the equation to solve for dy/dx.
Finally, the conclusion is that the derivative of the given equation with respect to x is (32x3 + 3√y) / (8y - 3xy(-1/2)).
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Consider the following quadratic function. f(x)=−2x^2 − 4x+1 (a) Write the equation in the form f(x)=a(x−h)^2 +k. Then give the vertex of its graph. (b) Graph the function. To do this, plot five points on the graph of the function: the vertex, two points to the left of the vertex, and two points to the right of the vertex. Then click on the graph-a-function button.
(a) In order to write the equation in the form f(x) = a(x - h)^2 + k, we need to complete the square and convert the given quadratic function into vertex form, where h and k are the coordinates of the vertex of the graph, and a is the vertical stretch or compression coefficient. f(x) = -2x² - 4x + 1
= -2(x² + 2x) + 1
= -2(x² + 2x + 1 - 1) + 1
= -2(x + 1)² + 3Therefore, the vertex of the graph is (-1, 3).
Thus, f(x) = -2(x + 1)² + 3. The vertex of its graph is (-1, 3). (b) To graph the function, we can first list the x-coordinates of the points we need to plot, which are the vertex (-1, 3), two points to the left of the vertex, and two points to the right of the vertex.
Let's choose x = -3, -2, -1, 0, and 1.Then, we can substitute each x value into the equation we derived in part
(a) When we plot these points on the coordinate plane and connect them with a smooth curve, we obtain the graph of the quadratic function. f(-3) = -2(-3 + 1)² + 3
= -2(4) + 3 = -5f(-2)
= -2(-2 + 1)² + 3
= -2(1) + 3 = 1f(-1)
= -2(-1 + 1)² + 3 = 3f(0)
= -2(0 + 1)² + 3 = 1f(1)
= -2(1 + 1)² + 3
= -13 Plotting these points and connecting them with a smooth curve, we get the graph of the quadratic function as shown below.
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for the solid, each cross section perpendicular to the x-axis is a rectangle whose height is three times its width in the xy-plane. what is the volume of the solid?
The volume of the solid can be found by integrating 3w² with respect to x, from the unknown limits of a to b.
To find the volume of the solid, we can use the concept of integration.
Let's assume the width of each rectangle is "w". According to the given information, the height of each rectangle is three times the width, so the height would be 3w.
Now, we need to find the limits of integration. Since the cross sections are perpendicular to the x-axis, we can consider the x-axis as the base. Let's assume the solid lies between x = a and x = b.
The volume of the solid can be calculated by integrating the area of each cross section from x = a to x = b.
The area of each cross section is given by:
Area = width * height
= w * 3w
= 3w²
Now, integrating the area from x = a to x = b gives us the volume of the solid:
Volume = [tex]\int\limits^a_b {3w^2} \, dx[/tex]
To find the limits of integration, we need to know the values of a and b.
In conclusion, the volume of the solid can be found by integrating 3w² with respect to x, from the unknown limits of a to b. Since we don't have the specific values of a and b, we cannot determine the exact volume of the solid.
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after you find the confidence interval, how do you compare it to a worldwide result
To compare a confidence interval obtained from a sample to a worldwide result, you would typically check if the worldwide result falls within the confidence interval.
A confidence interval is an estimate of the range within which a population parameter, such as a mean or proportion, is likely to fall. It is computed based on the data from a sample. The confidence interval provides a range of plausible values for the population parameter, taking into account the uncertainty associated with sampling variability.
To compare the confidence interval to a worldwide result, you would first determine the population parameter value that represents the worldwide result. For example, if you are comparing means, you would identify the mean value from the worldwide data.
Next, you check if the population parameter value falls within the confidence interval. If the population parameter value is within the confidence interval, it suggests that the sample result is consistent with the worldwide result. If the population parameter value is outside the confidence interval, it suggests that there may be a difference between the sample and the worldwide result.
It's important to note that the comparison between the confidence interval and the worldwide result is an inference based on probability. The confidence interval provides a range of values within which the population parameter is likely to fall, but it does not provide an absolute statement about whether the sample result is significantly different from the worldwide result. For a more conclusive comparison, further statistical tests may be required.
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Find the coordinates of the center of mass of the following solid with variable density. R={(x,y,z):0≤x≤8,0≤y≤5,0≤z≤1};rho(x,y,z)=2+x/3
The coordinates of the center of mass of the solid are (5.33, 2.5, 0.5).The center of mass of a solid with variable density is found by using the following formula:\bar{x} = \frac{\int_R \rho(x, y, z) x \, dV}{\int_R \rho(x, y, z) \, dV},
where R is the region of the solid, $\rho(x, y, z)$ is the density of the solid at the point (x, y, z), and dV is the volume element.
In this case, the region R is given by the set of points (x, y, z) such that 0 ≤ x ≤ 8, 0 ≤ y ≤ 5, and 0 ≤ z ≤ 1. The density of the solid is given by ρ(x, y, z) = 2 + x/3.
The integrals in the formula for the center of mass can be evaluated using the following double integrals:
```
\bar{x} = \frac{\int_0^8 \int_0^5 (2 + x/3) x \, dx \, dy}{\int_0^8 \int_0^5 (2 + x/3) \, dx \, dy},
```
```
\bar{y} = \frac{\int_0^8 \int_0^5 (2 + x/3) y \, dx \, dy}{\int_0^8 \int_0^5 (2 + x/3) \, dx \, dy},
\bar{z} = \frac{\int_0^8 \int_0^5 (2 + x/3) z \, dx \, dy}{\int_0^8 \int_0^5 (2 + x/3) \, dx \, dy}.
Evaluating these integrals, we get $\bar{x} = 5.33$, $\bar{y} = 2.5$, and $\bar{z} = 0.5$.
The center of mass of a solid is the point where all the mass of the solid is concentrated. It can be found by dividing the total mass of the solid by the volume of the solid.
In this case, the solid has a variable density. This means that the density of the solid changes from point to point. However, we can still find the center of mass of the solid by using the formula above.
The integrals in the formula for the center of mass can be evaluated using the change of variables technique. In this case, we can change the variables from (x, y) to (u, v), where u = x/3 and v = y. This will simplify the integrals and make them easier to evaluate.
After evaluating the integrals, we get $\bar{x} = 5.33$, $\bar{y} = 2.5$, and $\bar{z} = 0.5$. This means that the center of mass of the solid is at the point (5.33, 2.5, 0.5).
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Write an equation for the translation of y=6/x that has the asymtotes x=4 and y=5.
To write an equation for the translation of y = 6/x that has the asymptotes x = 4 and y = 5, we can start by considering the translation of the function.
1. Start with the original equation: y = 6/x
2. To translate the function, we need to make adjustments to the equation.
3. The asymptote x = 4 means that the graph will shift 4 units to the right.
4. To achieve this, we can replace x in the equation with (x - 4).
5. The equation becomes: y = 6/(x - 4)
6. The asymptote y = 5 means that the graph will shift 5 units up.
7. To achieve this, we can add 5 to the equation.
8. The equation becomes: y = 6/(x - 4) + 5
Therefore, the equation for the translation of y = 6/x that has the asymptotes x = 4 and y = 5 is y = 6/(x - 4) + 5.
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Now, the equation becomes y = 6/(x - 4).
To translate the equation vertically, we need to add or subtract a value from the equation. Since the asymptote is y = 5, we want to translate the equation 5 units upward. Therefore, we add 5 to the equation.
Now, the equation becomes y = 6/(x - 4) + 5.
So, the equation for the translation of y = 6/x with the asymptotes x = 4 and y = 5 is y = 6/(x - 4) + 5.
This equation represents a translated graph of the original function y = 6/x, where the graph has been shifted 4 units to the right and 5 units upward.
The given equation is y = 6/x. To translate this equation with the asymptotes x = 4 and y = 5, we can start by translating the equation horizontally and vertically.
To translate the equation horizontally, we need to replace x with (x - h), where h is the horizontal translation distance.
Since the asymptote is x = 4, we want to translate the equation 4 units to the right. Therefore, we substitute x with (x - 4) in the equation.
Now, the equation becomes y = 6/(x - 4).
To translate the equation vertically, we need to add or subtract a value from the equation.
Since the asymptote is y = 5, we want to translate the equation 5 units upward. Therefore, we add 5 to the equation.
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Find the derivative of p(t).
p(t) = (e^t)(t^3.14)
Therefore, the derivative of [tex]p(t) = (e^t)(t^{3.14})[/tex] is: [tex]p'(t) = e^t * t^{3.14} + 3.14 * e^t * t^2.14.[/tex]
To find the derivative of p(t), we can use the product rule and the chain rule.
Let's denote [tex]f(t) = e^t[/tex] and [tex]g(t) = t^{3.14}[/tex]
Using the product rule, the derivative of p(t) = f(t) * g(t) can be calculated as:
p'(t) = f'(t) * g(t) + f(t) * g'(t)
Now, let's find the derivatives of f(t) and g(t):
f'(t) = d/dt [tex](e^t)[/tex]
[tex]= e^t[/tex]
g'(t) = d/dt[tex](t^{3.14})[/tex]
[tex]= 3.14 * t^{(3.14 - 1)}[/tex]
[tex]= 3.14 * t^{2.14}[/tex]
Substituting these derivatives into the product rule formula, we have:
[tex]p'(t) = e^t * t^{3.14} + (e^t) * (3.14 * t^{2.14})[/tex]
Simplifying further, we can write:
[tex]p'(t) = e^t * t^{3.14} + 3.14 * e^t * t^{2.14}[/tex]
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