If a curve passes through the point (0,5) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. The equation of the curve is y = 5e^(²x).
How to find the equation?Let's assume that the equation of the curve is y = f(x), where f(x) is some unknown function.
We are given that the slope of the curve at every point P is twice the y-coordinate of P. This means that:
f'(x) = 2f(x)
where:
f'(x) is the derivative of f(x) with respect to x.
We can solve this differential equation by separating the variables and integrating:
1/f(x) df/dx = 2
Integrating both sides with respect to x, we get:
ln|f(x)| = 2x + C
where C is a constant of integration.
To find the value of C, we can use the fact that the curve passes through the point (0,5). Substituting x=0 and y=5 into the equation, we get:
ln|f(0)| = C
Since ln|f(0)| is just a constant, we can write it as another constant, say A. Therefore:
ln|f(x)| = 2x + A
Taking the exponential of both sides, we get:
|f(x)| = e^(²x+A)
Since f(x) cannot be negative (otherwise, the slope of the curve would be negative at some points), we can drop the absolute value signs:
f(x) = e^(²x+A)
To find the value of A, we can use the fact that the curve passes through the point (0,5). Substituting x=0 and y=5 into the equation, we get:
f(0) = e^A = 5
Therefore, A = ln(5). Substituting this value into the equation, we get:
f(x) = e^(²x+ln(5)) = 5e^(²x)
So the equation of the curve is y = 5e^(²x).
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10. Suppose y = x2 - 2x - 3. What is a linear equation that intersects the graph of
y=x²-2x-3 in exactly two places? Name the two points of intersection.
well, let's pick any two random x-values on the quadratic, hmmm say let's use x = 4 and x = 7, so hmm f(4) = 5 and f(7) = 32, that'd give us the points of (4, 5) and (7 , 32).
To get the equation of any straight line, we simply need two points off of it, let's use those two above.
[tex](\stackrel{x_1}{4}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{7}~,~\stackrel{y_2}{32}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{32}-\stackrel{y1}{5}}}{\underset{\textit{\large run}} {\underset{x_2}{7}-\underset{x_1}{4}}} \implies \cfrac{ 27 }{ 3 } \implies 9[/tex]
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{5}=\stackrel{m}{ 9}(x-\stackrel{x_1}{4}) \\\\\\ y-5=9x-36\implies {\Large \begin{array}{llll} y=9x-31 \end{array}}[/tex]
Check the picture below.
Refer to the attached images.
Compute the correlation coefficient for the following: X: -5, 1, 2, 11. Y: 5, 3, -3, 0
The correlation coefficient between X and Y is approximately -0.483.
We must first determine the mean and standard deviation for both X and Y in order to calculate the correlation coefficient between X and Y.
X has a mean of:
(-5 + 1 + 2 + 11)/4 = 2.25 is the mean of X.
Y has a mean of:
average of Y = (5 + 3 - 3 + 0)/4 = 1.
The value of X's standard deviation is
s_X = sqrt([(-5 - 2.25)^2 + (1 - 2.25)^2 + (2 - 2.25)^2 + (11 - 2.25)^2]/3) = 5.1478
The value of Y's standard deviation is
s_Y = sqrt([(5 - 1)^2 + (3 - 1)^2 + (-3 - 1)^2 + (0 - 1)^2]/3) = 3.7712
After that, we can figure out the correlation between X and Y:
cov(X, Y) = [(-5 - 2.25) × (5 - 1) + (1 - 2.25) × (3 - 1) + (2 - 2.25) × (-3 - 1) + (11 - 2.25) × (0 - 1)]/3 = -7.25
We can finally determine the correlation coefficient:
r = cov(X, Y)/(s X, s Y)=-7.25/(5.1478 3.7712)=-0.483
Thus, roughly -0.483 is the correlation coefficient between X and Y.
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3/4(8x+12)+x-5
this is hard can someone help
From the given expression, 3/4(8x+12)+x-5, the simplified expression is 7x + 4.
Simplifying an ExpressionFrom the question, we are to simplify the given expression.
To simplify the expression 3/4(8x+12)+x-5, we can follow the order of operations, which is commonly remembered using the acronym PEMDAS:
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
Using PEMDAS, we can simplify the expression as
3/4(8x+12)+x-5
Distributing 3/4 to the terms inside the parentheses
= 3/4 * 8x + 3/4 * 12 + x - 5
Simplifying the multiplication
= 6x + 9 + x - 5
Combining like terms
= 7x + 4
Hence, the simplified expression is 7x + 4.
Here is the complete question:
Simplify 3/4(8x+12)+x-5
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A rubber bouncy ball is dropped at a height of 120.00 inches onto a hard flat floor. After each bounce, the ball returns to a height that is 20\% less than the previous maximum height. What is the maximum height reached after the 7th bounce? Round your answer to the nearest hundredth.
Answer:
To solve the problem, we need to find the maximum height reached by the ball after the 7th bounce, given that each bounce has a rebound height of 20% less than the previous maximum height.
Let's start by finding the maximum height reached by the ball on the first bounce. The ball is dropped from a height of 120.00 inches, so the maximum height reached on the first bounce is:
120.00 inches
For each subsequent bounce, the maximum height reached is 20% less than the previous maximum height. We can express this mathematically as:
maximum height = 0.8 * previous maximum height
Using this formula, we can calculate the maximum height reached on the second bounce as:
maximum height on 2nd bounce = 0.8 * 120.00 inches = 96.00 inches
On the third bounce, the maximum height reached is 20% less than 96.00 inches:
maximum height on 3rd bounce = 0.8 * 96.00 inches = 76.80 inches
We can continue this pattern for each subsequent bounce. The maximum height reached on the fourth bounce is:
maximum height on 4th bounce = 0.8 * 76.80 inches = 61.44 inches
The maximum height reached on the fifth bounce is:
maximum height on 5th bounce = 0.8 * 61.44 inches = 49.15 inches
The maximum height reached on the sixth bounce is:
maximum height on 6th bounce = 0.8 * 49.15 inches = 39.32 inches
Finally, the maximum height reached on the seventh bounce is:
maximum height on 7th bounce = 0.8 * 39.32 inches = 31.46 inches
Therefore, the maximum height reached by the ball after the 7th bounce is approximately 31.46 inches. Rounded to the nearest hundredth, this is 31.45 inches.
Quadratic formula in standard form for points (2.5, 1.5), (2,0) and (3,0)
The Quadratic formula in standard form for points (2.5, 1.5), (2,0) and (3,0) are x²-4x + 3.75 = 0 and x² -2x = 0
What is the standard for for quadratic equation?The standard form of a quadratic equation is ax² + bx + c = 0, where 'a' is the leading coefficient and it is a non-zero real number
The given parameters that will help us to find the equations are
x = 2.5, and x = 1.5
(x-2.5) (x- 1.5)
Opening the brackets we have
x²-1.5x -2.5x + 3.75 = 0
x²-4x + 3.75 = 0
and the second is given as 2,0
x=2 and x=0
(x-2)(x-0)
Opening the brackets we have
x² -0 -2x+0 = 0
x² -2x = 0
Therefore the standard forms for the quadratic equations are x²-4x + 3.75 = 0 and x² -2x = 0
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Un móvil parte del reposo acelerando a razón de 3m/. Calcular el espacio recorrido en el tercer segundo
Answer:
El espacio recorrido en el tercer segundo será de 9 m.
Step-by-step explanation:
Mr. Phillips is mixing paint for his art class. How many 6-ounce bottles of paint can he fill with the quantities of pain. 64 ounces of blue
12 ounces of yellow
32 ounces
Mr. Phillips can fill 10 bottles of paint with the 64 ounces of blue paint and five bottles of paint with the 12 ounces of yellow paint, for a total of fifteen 6-ounce bottles.
He has 64 ounces of blue paint and 12 ounces of yellow paint. In order to figure out how many 6-ounce bottles of paint he can fill with the quantities of pain, he must divide the amount of paint by the size of the bottles. For the blue paint, he needs to divide 64 ounces by 6 ounces, which will equal 10.6 (rounded down to 10). For the yellow paint, he needs to divide 12 ounces by 6 ounces, which will equal 2 (rounded up to 3). This means that he can fill 10 bottles of paint with the 64 ounces of blue paint and three bottles of paint with the 12 ounces of yellow paint, for a total of thirteen 6-ounce bottles. The process of figuring out how many bottles of paint can be filled with the given amounts of paint is a simple one. It requires the painter to divide the amount of paint by the size of the bottles. Once the painter knows the answer, they can use it to accurately fill the appropriate number of bottles for their project. This is an important skill for a painter to have as it helps them plan for the necessary supplies and create a successful project.
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In the scale drawing of the base of a rectanglar swimming pool, the pool is 8.5 inches long and 4.5 inches wide. If 1 inch on the scale drawing is equivalent to 4 meters of actual length, what are the actual length and width of the swimming pool
Answer:
34 m x 18 m
Step-by-step explanation:
No picture shown, but we can do it anyway. If one inch on the drawing is actually 4 meters, multiply each dimension by 4.
The actual length is (8.5 * 4) = 34 meters.
The actual width is (4.5 * 4) = 18 meters.
The actual dimensions of the pool are 34 meters long and 18 meters wide.
X is a positive discrete uniform random variable. The mean and
the variance of X are equal to (µ,sigma2) = (5,4). Find
P(X>=6).
answers
(a) 3\7
(b) 1\2
(c) 2\7
As X is a discrete uniformly distributed random variable with a mean of 5, a variance of 4, and a range of values from 1, 2, 3, 5, 6, 7, 8, and 10, we know that X can have any one of these values with an equal probability of 1/10.
Calculating the likelihood that X will have a number equal to or greater to 6 is necessary to get [tex]P(X > =6)[/tex]. Due to the fact that X is a continuous uniformly distributed random variable.
We may calculate this probability simply counting all number of potential values for X that are greater or equal to to 6, dividing by the entirety of the potential values for X, and then adding the results together.
The probability of X is more than or equivalent to 6 is given by the fact there are a total of 5 variables of X that are at least equal to 6, namely: 6, 7, 8, 9, and 10.
[tex]P(X > =6) = 5/10 = 1/2[/tex]
Thus, (b) 1/2 is the right response.
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Q28) A neighborhood depanneur has determined that daily demand for milk cartons has an approximate normal distribution, with a mean of 65 cartons and a standard deviation of 7 cartons. On Saturdays, the demand for milk is known to exceed 71 cartons. On the coming Saturday, what is the probability that it will be at least 81 cartons?
The probability that there will be at least 81 cartoons on a coming Saturday is given as follows:
0.011 = 1.1%.
How to obtain probabilities using the normal distribution?The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.The mean and the standard deviation for this problem are given as follows:
[tex]\mu = 65, \sigma = 7[/tex]
The probability that there will be at least 81 cartoons is one subtracted by the p-value of Z when X = 81, hence:
Z = (81 - 65)/7
Z = 2.29
Z = 2.29 has a p-value of 0.989.
Hence:
1 - 0.989 = 0.011 = 1.1%.
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What is a quadratic function (f) whose zeros are -2 and 11
[tex]\begin{cases} x = -2 &\implies x +2=0\\ x = 11 &\implies x -11=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x +2 )( x -11 ) = \stackrel{0}{y}}\hspace{5em}\stackrel{\textit{now, assuming that}}{a=1}\hspace{3em}1( x +2 )( x -11 ) =y \\\\\\ ~\hfill {\Large \begin{array}{llll} x^2-9x-22=f(x) \end{array}} ~\hfill[/tex]
The diagram shows the graph of y = f(x) for -3.5 ≤ x ≤ 1.5
Find fg(-3)
From the graph and the given information, the value of the given function, fg(-3), is 9
Evaluating a functionFrom the question, we are to determine the value of the function, fg(-3).
First , we will determine the value of g(-3)
From the given information,
g(x) = 1/(2 + x)
Substitute x = -3 in the function to determine the value of g(-3)
g(-3) = 1/(2 + (-3))
g(-3) = 1/(2 - 3)
g(-3) = 1/-1
g(-3) = -1
Now,
fg(-3) = f(g(-3))
Thus,
fg(-3) = f(-1)
Now, we will determine the value of f(-1) from the given graph.
From the graph, we can read that the value of f(-1) is 9
Hence,
The value of fg(-3) is 9
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ABCE is a trapezium
AD is parallel to BC
DE = 4cm
Area of ABCE = 60cm^2
Area of ABCD = 48cm^2
Work out the values of f and g
The values of f and g that has been worked out is given as 6 and 8
How to solve the trapeziumThis is a question that would examine the combination of two shapes into one.
We have area of a triangle = base * height / 2
this would give
f * 4 / 2 = 60 - 48
2f = 12
divide through by 2
f = 12 / 2
f = 6
we have f * g = 48
given that the value of f = 6
g * 6 = 48
6g = 48
Divide through by 6 to get the value of f
6 g / 6 = 48 / 6
g = 8
Hence the value of f is given as 6 and the value of g is given as 8
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researchers are testing a new diagnostic tool designed to identify a certain condition. the null hypothesis of the significance test is that the diagnostic tool is not effective in detecting the condition. for the researchers, the more consequential error would be that the diagnostic tool is not effective, but the significance test indicated that it is effective. which of the following should the researchers do to avoid the more consequential error? responses increase the significance level to increase the probability of type i error. increase the significance level to increase the probability of type 1 error. increase the significance level to decrease the probability of type i error. increase the significance level to decrease the probability of type 1 error. decrease the significance level to increase the probability of type i error. decrease the significance level to increase the probability of type 1 error. decrease the significance level to decrease the probability of type i error. decrease the significance level to decrease the probability of type 1 error. decrease the significance level to decrease the standard error.
To decrease the probability of type I error in order to avoid the more consequential error of falsely concluding that the diagnostic tool is effective.
The researchers should decrease the significance level to decrease the probability of type I error. This is because a type I error occurs when the null hypothesis is rejected when it is actually true. In this case, the null hypothesis is that the diagnostic tool is not effective in detecting the condition.
If the researchers decrease the significance level, they are decreasing the probability of making a type I error, which means they are less likely to conclude that the diagnostic tool is effective when it is actually not. This will help the researchers avoid the more consequential error of falsely concluding that the diagnostic tool is effective.
In general, the significance level (also known as the alpha level) is the probability of making a type I error. By decreasing the significance level, the researchers are making the criteria for rejecting the null hypothesis more stringent, which reduces the likelihood of making a type I error.
It is important to note that decreasing the significance level will also decrease the power of the test, which is the probability of correctly rejecting the null hypothesis when it is false.
However, in this case, the researchers are more concerned with avoiding the more consequential error of falsely concluding that the diagnostic tool is effective, so decreasing the significance level is the appropriate action to take.
In conclusion, the researchers should decrease the significance level to decrease the probability of type I error in order to avoid the more consequential error of falsely concluding that the diagnostic tool is effective.
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For the following exercises, find the function if sint=x+1x. cost 44. sect cott 46. cos(sin−1(x+1x)) 47. tan−1(2x+1x) For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression. 16. cscxtanx+cotx;cosx 17. 1+tanxsecx+cscx;sinx 18. 1+sinxcosx+tanx;cosx 19. sinxcosx1−cotx;cotx For the following exercises, verify the identity. 29. cosx−cos3x=cosxsin2x 30. cosx(tanx−sec(−x))=sinx−1 31. cos2x1+sin2x=cos2x1+cos2xsin2x=1+2tan2x 32. (sinx+cosx)2=1+2sinxcosx 33. cos2x−tan2x=2−sin2x−sec2x For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression. 40. 1−tan2θcos2θ−sin2θ=sin2θ 41. 3sin2θ+4cos2θ=3+cos2θ 42. cotθ+cosθsecθ+tanθ=sec2θ
44. cos(x+1x)
46. sec(x+1x)
47. tan−1(2x+1x)
16.cscxcosx+cotx
17.1+sinx
18.1+cosx
19. cotx−1
29. True
30 True
31. True
32. True
33. True
40. False. The equivalent expression is sin2θ−1+tan2θcos2θ
41. False. The equivalent expression is 3sin2θ+4cos2θ−3
42. False. The equivalent expression is cotθ+cosθsecθ+tanθ−sec2θ
For the following exercises, find the function if sint=x+1x:
cos 44: cos(x+1x)sec cott 46: sec(x+1x)cos(sin−1(x+1x)) 47: cos(sin−1(x+1x))tan−1(2x+1x) 47: tan−1(2x+1x)For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression:
16. cscxtanx+cotx;cosx: cscxcosx+cotx17. 1+tanxsecx+cscx;sinx: 1+sinx18. 1+sinxcosx+tanx;cosx: 1+cosx19. sinxcosx1−cotx;cotx: cotx−1For the following exercises, verify the identity:
29. cosx−cos3x=cosxsin2x: True.30. cosx(tanx−sec(−x))=sinx−1: True.31. cos2x1+sin2x=cos2x1+cos2xsin2x=1+2tan2x: True.32. (sinx+cosx)2=1+2sinxcosx: True.33. cos2x−tan2x=2−sin2x−sec2x: True.For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression:
40. 1−tan2θcos2θ−sin2θ=sin2θ: False. The equivalent expression is sin2θ−1+tan2θcos2θ.41. 3sin2θ+4cos2θ=3+cos2θ: False. The equivalent expression is 3sin2θ+4cos2θ−3.42. cotθ+cosθsecθ+tanθ=sec2θ: False. The equivalent expression is cotθ+cosθsecθ+tanθ−sec2θ.Learn more about Trigonometry
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Write an equation for each line
slope=5/6;through(22,12)
The equation for the line with a slope of 5/6 and passing through the point (22,12) is y=5/6x - 16.
The equation for the line with a slope of 5/6 and passing through the point (22,12) can be expressed as y=5/6x + b. To calculate the value of b, we can plug in the given point's coordinates, (22,12), into the equation. This will result in the equation 12=5/6(22)+b. Solving for b, we get b=-16. Therefore, the equation for the line is y=5/6x-16.
In summary, we can express the equation for the line with a slope of 5/6 and passing through the point (22,12) as y=5/6x - 16. To calculate the value of b, we plugged the point's coordinates, (22,12), into the equation, resulting in the equation 12=5/6(22)+b. Solving for b, we got b=-16. Thus, the equation for the line is y=5/6x - 16.
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In circle H with m \angle GHJ= 90m∠GHJ=90 and GH=20GH=20 units, find the length of arc GJ.
HELP!!
The radius is equal to the length of the given line segment GH, which is 20 units. In this case, r = 20, and θ = 90° = π/2 radians.
Arc GJ is the portion of the circumference of circle H that lies between points G and J. Arc GJ can be calculated using the formula l = rθ, where l is the length of the arc, r is the radius of the circle, and θ is the angle subtended by the arc in radians. In this case, r = GH = 20 and θ = 90° = π/2 radians. Therefore, l = 20π/2 = 10π. Therefore, the length of arc GJ is 10π units.For example, if the radius of circle H is 10 units, l = 10π/2 = 5π. Thus, the length of arc GJ would be 5π units.To calculate arc GJ, we must first determine the radius of circle H. The radius is equal to the length of the given line segment GH, which is 20 units. We then use the formula l = rθ, where l is the length of the arc, r is the radius of the circle, and θ is the angle subtended by the arc in radians. In this case, r = 20, and θ = 90° = π/2 radians.
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What is the length of jg? 4 units 5 units 6 units 9 units
The length of JG is 5 units if ∠EDH ≅ ∠EDG. Thus option b is the correct answer as per the given relation.
The Given data is as follows:
∠EDH ≅ ∠EDG
Here in the question it is given that, ∠EDH is similar to ∠EDG
The corresponding sides of two triangles are congruent are similar to each other.
EH=9, EJ=4.
EH = EG
EG = 9 -------equation(1)
EG = EJ+JG
EG = 4+JG -------equation(2)
From equations 1 and 2, we get
9 = 4 + JG
9 - 4 = JG
JG = 5 units
Therefore, we can conclude that the length of JG is 5 units.
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The complete question is-
In circle D, ∠EDH ≅ ∠EDG. What is the length of JG?
a. 4 units
b. 5 units
c. 6 units
d. 9 units
Answer: 5
Step-by-step explanation:
∠EDH ≅ ∠EDG, EH=9, EJ=4.Corresponding sides of two triangles are congruent. (CPCTC) .... (1)From the given figure it is clear that ... (2)Using (1) and (2), we getSubtract 4 from both the sides.
In design and Analysis of Experiments, 8th edition, D.C Montgomery described an experiment that determined the effect of four different types of tips in a hardness tester on the observed hardness of a metal alloy. Four specimens of alloy were obtained, and each tip was tested once on each specimen, producing the following dataspecimentype of tip 1. 2. 3. 41. 9.3 9.4. 9.6. 10.02. 9.4 9.3 9.8. 9.93. 9.2. 9.4. 9.5. 9.74. 9.7. 9.6 10.0 10.2is there a difference in hardness measurements between the tipsuse fisher's LSD method to investigate specific differences between the tipsanalyze the residuals from this experiment
In conclusion, we find evidence of significant differences in hardness measurements between the four types of tips, based on the one-way ANOVA. The LSD method can be used to investigate specific differences between pairs of means.
The residuals appear to be randomly distributed and normally distributed, indicating that the assumptions of the ANOVA are reasonable.
To determine if there is a difference in hardness measurements between the tips, we can conduct a one-way ANOVA with the null hypothesis that the mean hardness measurements are equal for all four tips and the alternative hypothesis that at least one mean is different.
We can use Fisher's LSD method to investigate specific differences between the tips. The LSD method tests for significant differences between any pair of means using the formula:
LSD = tα(ν) × SE
where tα(ν) is the critical value of the t-distribution with α level of significance and (n-1) degrees of freedom, and SE is the standard error of the means, which is calculated as:
SE = √(MSE/n)
where MSE is the mean square error from the ANOVA and n is the sample size for each group.
First, we can perform the one-way ANOVA using software or a calculator. The results show that the F-statistic is significant at the 0.05 level with p-value < 0.05, which indicates that we reject the null hypothesis and conclude that there is at least one significant difference between the means.
Next, we can calculate the LSD for α = 0.05 and degrees of freedom (n-1) = 12 using the formula above. We obtain:
LSD = 2.179 × √(2.695/4) = 1.31
The LSD value indicates that if the difference between any two means is greater than 1.31, we can conclude with 95% confidence that the two means are significantly different.
To analyze the residuals, we can plot the residuals versus the fitted values and check for patterns or non-constant variance. We can also perform a normal probability plot of the residuals to check for normality.
The residual plot shows no clear patterns or trends, and the points appear to be randomly scattered around zero. The normal probability plot shows that the residuals follow a roughly straight line, indicating that they are approximately normally distributed.
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The area of the rectangle to the right is 18x² - 6x, and its width is 6x. Find the length of the rectangle.
Answer:
length = 3x - 1
Step-by-step explanation:
Area of a rectangle = length x width
or using l = length, w = width and A = area
A = l x w
or
l = A/w
Given
A = 18x² - 6x
w = 6x
l = 18x²- 6x/6x
= 18x²/6x - 6x/6x
= 3x - 1
Answer:
3x - 1
Step-by-step explanation:
Area of the Rectangle:
18x^2 - 6x
6x (3x - 1)
Width = 6x
Length =?
Area of Rectangle = Length * Width
Length = Area/breadth
6x (3x -1) / 6x cross out 6x
Thus, Length of Rectangle = 3x - 1
The point (2,3) lies on the line 2x+ky=19. Find the value of k.
Since the point (2, 3) lies on the line 2x + ky = 19, the value of k is equal to 5.
What is the slope-intercept form?Mathematically, the slope-intercept form of the equation of a straight line is given by this mathematical expression;
y = mx + c
Where:
m represents the slope or rate of change.x and y are the points.c represents the y-intercept or initial value.Making k the subject of formula, we have the following:
k = (19 - 2x)/y
Since the point (2, 3) lies on the line, we have:
k = (19 - 2(2))/3
k = (19 - 4)/3
k = 15/3
k = 5.
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A cone of radius rcm and height 3rcm is removed from a cone of radius 10 cm and height 30 cm to give a frustum. The volume of the frustum is 2855 cm³ Calculate the value of r. Show all your working.
If the volume of the frustum is 2855 cm³ and the value of r = 28.4
What is a cone?A cone is a three-dimensional shape in geometry that narrows smoothly from a flat base (usually circular base) to a point called the apex or vertex.
Volume of the = (1/3*п*R²*H) + 1/3*п*r²*h
2855 = 1/3*22/7*10*30 + 1/3*22/7*r²*3
2855 = 314.3 + 66r²/21
2540.7 *21= 66r²
53354.7 = 66r²
r² = 808.4
r √808.4
Therefore the value of r = 28.4
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Write an exponential function that passes through (0,3) and (1,6). Write your answer in the form f(x) = ab^x
The exponential function that passes through (0,3) and (1,6) is [tex]f(x) = 3(2^{x})[/tex]
What is the exponential function?An exponential function is a mathematical function in which an independent variable appears in the exponent. In other words, the function has the form f(x) = [tex]a^{x}[/tex], where 'a' is a constant greater than 0 and not equal to 1, and 'x' is the independent variable.
To find the exponential function that passes through (0,3) and (1,6), we need to determine the values of 'a' and 'b' in the function [tex]f(x)=a(b^{x})[/tex].
First, we can use the point (0,3) to find the value of 'a'. Plugging in x=0 and y=3, we get:
3 = [tex]ab^0[/tex]
3 = a
Next, we can use the point (1,6) to find the value of 'b'. Plugging in x=1 and y=6, we get:
6 = [tex]ab^1[/tex]
6 = 3b
b = 2
Now that we have the values of 'a' and 'b', we can write the exponential function in the form [tex]f(x)=a(b^{x})[/tex]. Substituting 'a' and 'b' into the equation, we get: [tex]f(x) = 3(2^{x})[/tex]
Therefore, the exponential function that passes through (0,3) and (1,6) is [tex]f(x) = 3(2^{x})[/tex]
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Can someone help me with this? I dont know whats its trying to ask. It decreases by 1/2 every year. URGENT
Answer:
Below
Step-by-step explanation:
Year 0 = 800
Year 1 = 400
Year 2 = 200
Year 3 = 100
Year 4 = 50
Year 5 = 25
From year 2 to year 4 = 150 dollars decrease in 2 years = 75 dollars per year
this is just a quick addition to "jsimpson11000" good reply above
so we know the decrease is exponential, that means we have an equation about V = abᵗ, now, hmmm who knows what "ab" is, now, once we know that, then we can get the "slope" from t=2 to t=4, so let's use the table to get it.
[tex]{\Large \begin{array}{llll} V=ab^t \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} t=3\\ V=100 \end{cases}\implies 100=ab^3 \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} t=5\\ V=25 \end{cases}\implies 25=ab^5\implies 25=ab^3b^2\implies \stackrel{\textit{substituting from above}}{25=(100)b^2} \\\\\\ \cfrac{25}{100}=b^2\implies \cfrac{1}{4}=b^2\implies \sqrt{\cfrac{1}{4}}=b\implies \boxed{\cfrac{1}{2}=b} \\\\[-0.35em] ~\dotfill[/tex]
[tex]100=ab^3\implies 100=a\left( \cfrac{1}{2} \right)^3\implies 100=\cfrac{a}{8}\implies \boxed{800=a} \\\\\\ ~\hfill {\Large \begin{array}{llll} V=800\left( \frac{1}{2} \right)^t \end{array}}~\hfill[/tex]
now let's find the slope
[tex]\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill\\\\ V(t)= 800\left( \frac{1}{2} \right)^t\qquad \begin{cases} t_1=2\\ t_2=4 \end{cases}\implies \cfrac{V(4)-V(2)}{4 - 2} \\\\\\ \cfrac{(50)~~ - ~~(200)}{2}\implies \text{\LARGE -75}[/tex]
so is a negative slope, because is Decay or decrement, however you're expected to enter it as positive, so in essence just the absolute value change.
In one lottery, a player wins the jackpot by matching all five distinct numbers drawn in any order from the white balls (1 through ) and matching the number on the gold ball (1 through ). If one ticket is purchased, what is the probability of winning the jackpot?
Answer: The total number of possible outcomes in the lottery can be calculated by finding the number of ways to choose 5 distinct numbers out of 70, and then multiply that by the number of possible choices for the gold ball (1 out of 25). So:
Total number of outcomes = (70 choose 5) * 25 = 25,989,600
To win the jackpot, the player must match all 5 white ball numbers and the gold ball number. The number of ways to do this is simply 1, since there is only one winning combination.
Therefore, the probability of winning the jackpot with a single ticket is:
Probability of winning = (number of winning outcomes) / (total number of outcomes) = 1 / 25,989,600
So the probability of winning the jackpot with a single ticket is approximately 0.00000003846, or 1 in 25,989,600.
Step-by-step explanation:
Joe ran 10 kilometers in 5 hours. what is joe’s unit rate
Answer:
2 kph
Step-by-step explanation:
10 kilometers in 5 hours.
1 hour = ?
10/5 = 2 Kilometers
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What is the length of x in the diagram below?
A triangle with vertical bisector forms 2 triangles with a right angle. One triangle has a side length of 5 and another angle with a measure 45 degrees. Another triangle has a hypotenuse with length x and another angle with measure 30 degrees.
StartFraction 5 Over StartRoot 3 EndRoot
StartFraction 10 Over StartRoot 3 EndRoot EndFraction
5 StartRoot 3 EndRoot
10
Answer:
The length of x is (10√3)/3.
Step-by-step explanation:
Using the trigonometric ratios of a 30-60-90 triangle, we know that the side opposite the 30-degree angle is half the length of the hypotenuse. Therefore, we have:
x/2 = (5/√3)
Multiplying both sides by 2, we get:
x = 10/√3, which can be simplified to:
x = (10√3)/3
So the length of x is (10√3)/3.
Hope this helps you! I'm sorry if it's wrong. If you need more help, ask me! :]
Find the measure of x. 8 62° X x = [?]
Answer:
x ≈ 17
Step-by-step explanation:
using the cosine ratio in the right triangle
cos62° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{8}{x}[/tex] ( multiply both sides by x )
x × cos62° = 8 ( divide both sides by cos62° )
x = [tex]\frac{8}{cos62}[/tex] ≈ 17 ( to the nearest whole number )
Which equation forms a line that is perpendicular to y = 3x + 4?
A. y = 3x + 6
B. y = -3x + 4
C. y = -1/3x + 2
Answer:
The answer will be B. y= -3x +6
The cuboid below is made of nickel and has a mass of 534 g. Calculate its density, in g/cm³. If your answer is a decimal, give it to 1 d.p
The sοlutiοn οf the given prοblem οf cubοid cοmes οut tο be the cubοid has a density οf abοut 6.0 g/cm³.
Describe the cubοid.In geοmetry, a hypercube is a cοncrete οr muti shape. A hypercube is a cοnvex pοlygοns with 12 sides, 6 rectangular faces, and 8 edges. A cubοid is anοther term fοr the creature. An οbject with six square sides is called a cube. Bricks and literature are amοng the items fοund in bοxes. The main differences between cubic but alsο cubic are as fοllοws.
Here,
This equatiοn can be changed tο read density = mass/vοlume. Since we already knοw the mass (534 g), we can use the rearranged methοd tο determine the vοlume.
=> Vοlume = L, W, and H, and mass = 534 g.
=> density equals mass/vοlume
=> weight = 534/ (L x W x H)
The CRC Encyclοpedia οf Chemistry and Physics states that nickel has a density οf 8.908 g/cm³.
Thus, we can enter this number as a substitute in οur fοrmula tο οbtain:
=> density = 534/(L, W, H)
=> 8.908 g/cm³.
Tο sοlve fοr density, we can rewrite this equatiοn as fοllοws:
=> density = 534/(L, W, H)
=> 6.0 g/cm³ (tο 1 decimal place)
Cοnsequently, the cubοid has a density οf abοut 6.0 g/cm³.
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