The complex fifth roots of 3 are Z₀ = ∛3 (cos 0° + i sin 0°), Z₁ = ∛3 (cos 72° + i sin 72°), Z₂ = ∛3 (cos 144° + i sin 144°), Z₃ = ∛3 (cos 216° + i sin 216°) and Z₄ = ∛3 (cos 288° + i sin 288°)
The complex fifth roots of 3 can be found by finding the roots of the equation z^5 = 3. To find the roots in polar form, we first convert 3 to polar form:
3 = |3| (cos 0 + i sin 0) = 3 (cos 0 + i sin 0)
Next, we find the argument of the fifth roots, which is equal to 360°/5 = 72°. The polar form of the first root is then:
z₁ = ∛3 (cos 0° + i sin 0°) = ∛3 (cos 0° + i sin 0°)
The polar form of the other roots can be found by rotating the argument by 72°, 144°, 216°, and 288° respectively. The result is:
z₂ = ∛3 (cos 72° + i sin 72°)
z₃ = ∛3 (cos 144° + i sin 144°)
z₄ = ∛3 (cos 216° + i sin 216°)
z₅ = ∛3 (cos 288° + i sin 288°)
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Derek tried to solve an equation step by step. 8d=72 8d/8 = 72/8 d=8 Find Derek's mistake
The mistake of Derek is that 72 divided by 8 is 9, not 8.
What is an Equation?An equation is the statement of two expressions located on two sides connected with an equal to sign. The two sides of an equation is usually called as left hand side and right hand side
Given is an equation that Derek is trying to solve.
The equation is 8d = 72.
8d = 72
In the first step, Derek divided both sides of the equation by 8, which is a correct way to solve the problem.
We have to divide both sides of the equation by 8.
8d / 8 = 72 / 8
This step is correct.
After dividing, in the left hand side, there will be only 'd' remaining and in the right hand side, 27/8 = 9.
d = 9
But Derek got 8 instead of 9 in the right hand side.
Hence the value of d is 9 and not 8, which is the mistake of Derek.
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determine whether the quantitative variable is discrete or continuous amount of water in a dogs bowl
The quantitative variable is continuous for any given value of the amount of water in a dog's bowl.
In order to give an answer to this question, we need to recall the definition of continuous and discrete variables. Then according to the definition, we get the answer if the quantitative variable is discrete or continuous for the given amount of water in a dog's bowl.
As we understand that a discrete variable is a variable whose value is fetched by counting. A discrete random variable holds countable possible values. A random variable is one whose value is numerical. On the other hand, a continuous variable is a variable whose value is fetched by measuring.
The amount of water in a dog's bowl has any value. So the amount of water in a dog's bowl is a continuous variable.
Hence, the quantitative variable is continuous for any given amount of water in a dog's bowl.
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Dorothy bought 4 boxes of chocolates for $12. Find the unit price.
Answer:
3
Step-by-step explanation:
12/4=3
given the experimental data, what is the activation energy, ea, (in kj/mol) for this reaction? hint: r
The activation energy for the given reaction is166 KJ/mol. (Option B)
Activation energy refers to the minimum amount of extra energy that is required by a reacting molecule to be converted into product. It is the minimum amount of energy required to energize or activate atoms or molecules so that they can undergo a chemical reaction or transformation. In order to calculate the activation energy (Ea), the Arrhenius equation is used which is given by:
k=Ae^((-E)/RT)
where k is the rate of chemical reaction, A is a constant depending on the chemicals involved, E is the activation energy, R is the universal gas constant, and T is the temperature.
The equation can be expressed as
ln〖k=ln〖A-(E/RT)lne〗 〗
2.303 log〖k=2.303 log〖A-E/RT〗 〗
log〖k=log〖A-E/2.303RT〗 〗
As there are two temperatures and two rate constants given, the Arrhenius equation is rearranged:
log〖k2/k1=E/2.303(1/T1〗-1/T2)
According to the data,
k1 = 2.7 x 10^-4
T1 = 600 K
k2 = 3.5 x 10^-3
T2 = 650 K
R = 8.314
log〖(3.5 x10^(-3))/(2.7 x 10^(-4) )=E/(2.303*8.314)(1/600〗-1/650)
E = 166208 J/mol or 166 KJ
Note: The question is incomplete. The complete question probably is: Calculate the activation energy (Ea) in kJ/mol for a reaction if the rate constant (k) is 2.7 × 10^-4 (M^−1sec^−1) at 600 K and 3.5 × 10^−3 (M−^1sec^−1) at 650 K. a. −166 kJ/mol b. 166 kJ/mol c. 1.64 kJ/mol d. 1.66 × 10^5 kJ/mol. (R
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The graph below is an example of which type of function?
O quadratic
O linear
O cubic
The graph below is an example of graph function is cubic.
What is graph function?We must identify the kind of the graph after receiving it.
A parabola represents the graph of a quadratic function. As a result, the presented graph is not a quadratic function graph.
A straight line is represented by a linear function. The presented graph is not a graph of a linear function as a result.
A square root function's graph resembles a half parabola. Therefore, it can't be the square root function graph.
A cubic function's final behaviour is "opposite at both ends." We can see that the provided graph satisfies this requirement.
As a result, the shown graph illustrates a cubic function.
The right graph is C.
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what is 5 3/5 divided by 2 2/3
Answer:
Exact Form:
21/10
Decimal Form:
2.1
Mixed Number Form:
2 1/10
Step-by-step explanation:
When adding two vectors, I should do the following: O Pick an angle and just go with it. O Write each vector in component form and add the components separately. O Combine all the vector components into one number and add them. O Calculate the magnitude of each vector and add them.
Write each vector in component form and add the components separately. This new vector will represent the sum of the two original vectors. This is the most straightforward way to add two vectors.
1. Write each vector in component form. This means breaking down each vector into its x- and y-components.
2. Add the x-components of each vector together and the y-components of each vector together.
3. The result will be a single vector with both x- and y-components.
To add two vectors, I should write each vector in component form, breaking them down into their x and y components. Then, I should add the components together, resulting in a new vector with components that are the sum of the two original vectors' components. This new vector will represent the sum of the two original vectors. This is the most way to add two vectors.
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Solve the equation for all real values of x.
3 tan²x =√3 tanx
The solution of the equation for all real values of x is πk, π/6.
How to solve the equation for all real values of x?Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, specifically right triangles.
3 tan²x = √3 tan x
Divide both sides by 3tan x:
tan x = (√3)/3
x = tan⁻¹((√3)/3)
x = 30°
x = π/6 radian
Note: multiply 30° by π/180 to convert to radian. 180° = π radian
Since π/6 (i.e. 30°) falls between 0 and 180°.
Thus, the solution of the equation for all real values of x is πk, π/6. We can choose πk, π/6.
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In a basketball game,Benito score 3 points less than twice the number of points that Carnell scored. Benito scored 13 points how many points did Carnell score
In a basketball game,Benito score 3 points less than twice the number of points that Carnell scored. Benito scored 13 points therefore the number of points Carnell scored was 16 points.
To get Carnell pointsBenito scored 13 points which means that the number of points Carnell scored was:
8 multiplied by 2 = 16
When we subtract 3 from 16 we get 13 which is Benito score mentioned in the question thereby making it the correct choice.
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One Step Equations Exit Ticket help i ned answers fast
The value of the equations are
n = 6
x = 20
b = -4
k = 176
What is an Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the equation be represented as A
Now , the value of A is
a)
-15 + n = -9 be equation (1)
On simplifying the equation , we get
Adding 15 on both sides of the equation , we get
n = -9 + 15
n = 6
Therefore , the value of n is 6
b)
x - 7 = 13 be equation (2)
On simplifying the equation , we get
Adding 7 on both sides of the equation , we get
x = 13 + 7
x = 20
Therefore , the value of x is 20
c)
14b = -56 be equation (3)
On simplifying the equation , we get
Dividing 14 on both sides of the equation , we get
b = - ( -56/14 )
b = -4
Therefore , the value of b is -4
d)
16 = k/11 be equation (4)
On simplifying the equation , we get
Multiply by 11 on both sides of the equation , we get
k = 16 x 11
k = 176
Therefore , the value of k is 176
Hence , the equations are solved
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Mr. Casey needs to drive 278 miles in 7 hours. He estimates that he should drive at a speed of 50 miles per hour. Is his estimate greater than or less than the actual speed at which he should travel? Explain.
The actual speed of Mr Casey is 39.71 miles per hour and his estimation is greater than the actual speed at which he should travel.
What is estimation?Estimation is a roughly calculated answer of something for which we get an approximate result of it.
Given that, Mr. Casey needs to drive 278 miles in 7 hours. He estimates that he should drive at a speed of 50 miles per hour. We need to find whether his estimation correct or not,
We know that,
Speed = distance / time
Therefore, Mr. Casey's actual speed =
Speed = 278 / 7
= 39.71 miles per hour
As, he estimated his speed being 5o miles per hour, and actual speed is 39.71 miles per hour, therefore, his estimation is greater than the actual speed at which he should travel.
Hence, the actual speed of Mr Casey is 39.71 miles per hour and his estimation is greater than the actual speed at which he should travel.
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Mr. Smith owns a car that is currently worth $8,000. The value of the car depreciates at the rate of 25% each year. Which of the following graphs represents the value of the car, y, in dollars, after x years? V
Answer:
Car Value Depreciation Graph
The graph that represents the value of the car, y, in dollars, after x years is the graph that starts at $8,000 on the y-axis and decreases at a steady rate of 25% each year on the x-axis. It would be the graph V.
3x+12y=-8 show ur work
Here is the attached image for the answer. If you write it in in slope-intercept form, y=mx+b. You will get the answer below in the attached image.
Use the distributive property to solve 12x9
The value of given expression 12*9 is 108
What is Algebraic expression ?
Algebraic expressions are the concept of expressing numbers the usage of letters or alphabets without specifying their real values. The basics of algebra taught us how to express an unknown fee using letters which includes x, y, z, and so forth. these letters are known as here as variables. An algebraic expression can be a aggregate of both variables and constants. Any price that is placed before and accelerated through a variable is a coefficient.
Given expression ,
12*9 = (3*4) * (3*3)
= 9*(3*4)
= 9* 12
= 108
Hence, The value of given expression 12*9 is 108
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4. Explain how to use division to find an
equivalent fraction for 20/16
Answer:
5/4
Step-by-step explanation:
divide 20 and 16 by 4
20/4=5
16/4=4
Housing prices in a small town are normally distributed with a mean of
$147,000 and a standard deviation of $7,000. Use the empirical rule to
complete the following statement.
Approximately 95% of housing prices are between a low price of
$Ex: and a high price of $
Approximately 95% of housing prices are between a low price of $133,000 and a high price of $161,000.
What is defined by the Empirical Rule?These approximate percentages are defined by the Empirical Rule, for a normally distributed variable.
68% of the measures within one standard deviation of the mean.95% of the measures within two standard deviation of the mean.99.7% of the measures within three standard deviation of the mean.Hence the prices within two standard deviations of the mean are given as follows:
147,000 - 2 x 7,000 = 133,000.147,000 + 2 x 7,000 = 161,000.More can be learned about the Empirical Rule at brainly.com/question/10093236
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5. Jenna is making
bracelets.
She needs 7
inches of string for each
bracelet. She has a spool
with 16 feet of string. How
many bracelets can Jenna
make?
Answer:
Step-by-step explanation:
Each bracelet need 7 inches of string. Jenna has a total of 16 feet of string.
Divide the total amount of string by the number of inches needed for each bracelet.
Step 1: Convert 16 feet into inches. 12 inches = 1 foot
16 feet x (12 inches/1 foot) = 192 inches
This is the total inches of string.
Note that feet and foot units cancel and you are left with inches.
Step 2: Find the number of bracelets that can be made. Use the answer from step 1. Each bracelet needs 7 inches of string.
Total inches of string / 7 inches of string for each bracelet
192 inches / 7 inches = 27.4
Round the answer to 27.
William and his children went into a restaurant that sells hamburgers for $8 each and drinks for $2.50 each. William has $95 to spend and must buy no less than 18 hamburgers and drinks altogether. If xx represents the number of hamburgers purchased and yy represents the number of drinks purchased, write and solve a system of inequalities graphically and determine one possible solution.
Answer: We can set up the following system of inequalities to represent the situation:
8x + 2.5y <= 95 (the total cost of the hamburgers and drinks must be less than or equal to $95)
x >= 18-y (William must buy no less than 18 hamburgers and drinks altogether)
x >= 0, y >= 0 (non-negative values for the number of hamburgers and drinks purchased)
To graph the system of inequalities, we can start by graphing the inequality 8x + 2.5y <= 95, which represents a boundary line in the xy-plane. Since it's a less or equal sign, we will shade the area that is below the line.
The second inequality represents a boundary line which is the line x = 18-y. Since x must be greater than or equal to 18-y, we will shade the area that is on or above the line.
The third inequality is a non-negative constraint for both x and y, so we will graph the x and y axis and shade the area that is above them.
After combining all the shaded areas, we'll get a feasible solution region that represents possible values of x and y that satisfies the system of inequalities.
One possible solution to the system of inequalities is x=24 and y=6, which is a point in the feasible region that satisfies all the inequalities.
It's worth noting that there could be other solutions as well, depending on the number of hamburgers and drinks William wants to buy.
Step-by-step explanation:
the manager of a large company that sells pet supplies online wants to increase sales by encouraging repeat purchases. the manager believes that if past customers are offered $10 off their next purchase, more than 40 percent of them will place an order. to investigate the belief, 90 customers who placed an order in the past year are selected at random. each of the selected customers is sent an e-mail with a coupon for $10 off the next purchase if the order is placed within 30 days. of those who receive the coupon, 38 place an order.
At the significance level of a = 0.05, there is no strong statistical support for the manager's assertion.
Accepting it would be a type II error because the null hypothesis might not be true.
The 'null' and 'alternative' hypotheses should be
p 0.4 for H0 and p > 0.4 for Ha
q= 1-p= 1-0.4= 0.6
The alpha level for significance is 0.05.
For one-tailed tests, the crucial area is Z> ± 1.645
The sample proportion is p^= x/n= 38/90=0.42222
Using the z statistic
z= p^- p/ √pq
z= 0.422-0.4/ √0.4*0.6
z= 0.04536
We are unable to reject the null hypothesis since the calculated value of z=0.04536 does not fall within the crucial zone Z> 1.645.
The manager's assertion is not sufficiently supported by the available data.
When we reject the actual null hypothesis, we commit a type I mistake.
When we accept the incorrect null hypothesis, we make type II errors.
Accepting it would be a type II error because the null hypothesis might not be true.
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-8.8/x = 1.5/10
Pls help
Answer:
x = 58.67
Step-by-step explanation:
[tex]\frac{-8.8}{x}[/tex] = [tex]\frac{1.5}{10}[/tex]
Cross multiply
1.5x = (-8.8)(10)
1.5x = -88
Divide, solve for x
x = -88/1.5 = -58.6666667 = around -58.67
Which of the following matrix multiplications are possible?
[a b] x [c d]
The matrix multiplication is possible for option (b) 1 × 3 by 3 × 3, and option (c) 2 × 1 by 1 × 2.
What is a matrix?
A matrix is a rectangular array or table with numbers or other objects arranged in rows and columns. Matrices is the plural version of matrix. The number of columns and rows is unlimited. Matrix operations include addition, scalar multiplication, multiplication, transposition, and many others.
It is known that if the number of columns of a matrix A be equal to the number of rows of another matrix B then the product of matrix A and matrix B i.e. AB is defined.
a) In this option no of columns in first matrix is 1 and the no of rows in second matrix is 2.
That means no of columns in first matrix is not equal to no of rows in second matrix.
Hence, in this case matrix multiplication is not possible.
Therefore, option (a) is incorrect.
b) In this option no of columns in first matrix is 3 and the no of rows in second matrix is also 3.
That means no of columns in first matrix is equal to no of rows in second matrix.
Hence, in this case matrix multiplication is possible.
Therefore, option (b) is correct.
c) In this option no of columns in first matrix is 1 and the no of rows in second matrix is 1.
That means no of columns in first matrix is equal to no of rows in second matrix.
Hence, in this case matrix multiplication is possible.
Therefore, option (c) is correct.
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Two angles are supplementary. If the mZA is five times the sum of the mZB plus 7.2°, what is mZB?
Due 01.
The value of angle mZB is 28. 8 degrees
How to determine the anglesIt is important to note that supplementary angles are described as pair of angles that sum up to angles on a straight line.
Also, angles on a straight is equal or equivalent to 180 degrees.
From the information given, we have that;
mZB+ mZA = 180mZA = 5mZB + 7. 2 degreesNow, susbtitute the values, we get;
mZB + mZA = 180
5mZB + 7. 2 + mZB = 180
collect like terms, we get;
5mZB + mZB = 180 - 7.2
Add or subtract the like terms
6mZB = 172.8
Now, divide both sides by the coefficient of mZB, we get;
mZB = 172.8/6
Divide the values
mZB = 28. 8 degrees
Hence, the value is 28. 8 degrees
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Sample Spaces and Probabilities (18 points) For each of the following scenarios first describe the sample space and indicate how big it is (i.e., what its cardinality is) and then answer the question. (a) [3 Points] You flip a fair coin 50 times. What is the probability of exactly 20 heads? (b) [3 Points] You roll 2 fair 6-sided dice, one red and one blue. What is the probability that the sum of the two values showing is 4? (c) [3 Points] You are given a random 5 card poker hand (selected from a single deck). What is the probability you have a full-house (3 cards of one rank and 2 cards of another rank)? (d) [3 Points] 20 labeled balls are placed into 10 labeled bins (with each placement equally likely). What is the probability that bin 1 contains exactly 3 balls? (e) [3 Points] There are 30 psychiatrists and 24 psychologists attending a certain conference. Three of these 54 people are randomly chosen to take part in a panel discussion. What is the probability that at least one psychologist is chosen? What is the probability that exactly three psychologists are chosen? (f) [3 Points] You buy ten cupcakes choosing from 3 different types (chocolate, vanilla and caramel). Cupcakes of the same type are indistinguishable. What is the probability that you have at least one of each type?
(a) Sample space: 2^50 (1,024,000). Probability of exactly 20 heads: 0.000244140625.
(b) 6^2 (36). Probability of a sum of 4: 1/9.
(c) C(52,5) (2,598,960). Probability of a full house: 0.001441.
(d) 10^20 (10^19). Probability of bin 1 containing exactly 3 balls: 1/10^19.
(e) C(54,3) (19,323). Probability of at least one psychologist: C(24,1)/C(54,3) (0.841). Probability of exactly three psychologists: C(24,3)/C(54,3) (0.037).
(f) 3^10 (590,490). Probability of at least one of each type: 0.001690.
(a) The sample space consists of all the possible outcomes of flipping a fair coin 50 times, which is 2^50 (1,024,000). There are only two possible outcomes for each flip, heads or tails. The probability of getting exactly 20 heads is 0.000244140625.
(b) The sample space consists of all the possible outcomes of rolling a red and a blue 6-sided die, which is 6^2 (36). The probability of getting a sum of 4 is 1/9.
(c) The sample space consists of all the possible 5 card poker hands (selected from a single deck), which is C(52,5) (2,598,960). The probability of getting a full house (3 cards of one rank and 2 cards of another rank) is 0.001441.
(d) The sample space consists of all the possible ways of placing 20 labeled balls into 10 labeled bins, which is 10^20 (10^19). The probability of bin 1 containing exactly 3 balls is 1/10^19.
(e) The sample space consists of all the possible ways of selecting 3 of the 54 people attending the conference, which is C(54,3) (19,323). The probability of at least one psychologist being chosen is C(24,1)/C(54,3) (0.841). The probability of exactly three psychologists being chosen is C(24,3)/C(54,3) (0.037).
(f) The sample space consists of all the possible ways of buying ten cupcakes from the 3 different types, which is 3^10 (590,490). The probability that you have at least one of each type is 0.001690.
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Draw two rectangles that have the same area but are not congruent.
Violet creates a rectangle with an area of 12 square feet. Bertie makes a rectangle with an area of 12 squares meters. Do the two rectangles have the same area?
Same-sized rectangles are possible. However, because their sides might differ in length, they are not congruent.
What does it mean when two rectangles are congruent?Congruent refers to two rectangles having the same dimensions and shape. So, two rectangles are said to be congruent if their length and width are equivalent in size. The length and width of two rectangles must match for them to be congruent.
Also having the same area are congruent shapes, such as rectangles. However, congruent rectangles with the same area are not always the case. There are differences among rectangles. Figures that have the exact same shape but differ in size are referred to as similar figures.
These two polygons are incongruent because while their matching sides are equal, their corresponding angles are not. Despite the sides being the same size, they have different forms.
The complete question is:
Draw two rectangles that have the same area but are not congruent.
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The town of Simpsonville has two tow truck companies. Company A charges an initial fee of $30 plus $2 per mile. Company B charges an initial fee of $20 plus $4 per mile. Use the graph to determine when it's cheaper to use Company B instead of Company A.
Using the graph, it's cheaper to use Company B instead of Company A; when the number of miles is less than 5 miles.
How to interpret Linear equation graphs?For us to solve this question, we need to first create an equation;
If A company initially supplies $30 and then $2 per mile, then the equation here is:
y = 2x + 30
If B company initially supplies $20 and then $4 per mile, then the equation is expressed as:
y = 4x + 20
Due to the fact that the two wages are asked to be equalized, we equate these two equations with each other and find the result.
2x + 30 = 4x + 20
2x = 10
x = 10/2
x = 5
This is the number of miles that both companies will charge a customer the same amount.
Then the time for which it is cheaper to use company B is when number of miles is less than 5.
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Find the surface area of each pyramid. Round to the nearest tenth if necessary.
triangular pyramid: base side lengths, 6 yd; base height: 5.2 yd; slant height, 11.7 yd
a.
136.5 yd2
b.
90.7 yd2
c.
226.2 yd2
d.
120.9 yd2
The total surface area of the triangular pyramid will be 106.86 yd².
What is the surface area of triangular pyramid?The formula to calculate the total surface area of a triangular pyramid is -
T.S.A. = [tex]\frac{1}{2}[/tex](a × b) + [tex]\frac{3}{2}[/tex](b × s).
Given is a triangular pyramid with dimensions -
base side lengths : 6 yd base height : 5.2 ydslant height : 11.7 ydThe total surface area of the triangular prism will be -
T.S.A. = [tex]\frac{1}{2}[/tex](a × b) + [tex]\frac{3}{2}[/tex](b × s).
T.S.A. = 1/2(5.2 x 6) + 3/2(5.2 x 11.7)
T.S.A. = 31.2/2 + 182.52/2
T.S.A. = 15.6 + 91.26
T.S.A. = 106.86 yd²
Therefore, the total surface area of the triangular pyramid will be 106.86 yd².
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Please help me out. I Need it today
Answer:
greetings!!!
Look at the attachment above
and If you have any questions or unclear ideas tag on the comments.
Hope it helps!!!
Qasim is deciding between two truck rental companies. Company A charges an initial fee of $15 for the rental plus $1 per mile driven. Company B charges an initial fee of $45 for the rental plus $0.50 per mile driven. Graph each function and determine which company would be cheaper if Qasim needs to drive 80 miles with the rented truck.
Equation for the cost of company A: f(x) = 15 + x
Equation for the cost of company B: g(x) = 45 + x/2
For 80 miles, company B is cheaper than company A.
What is linear equation?A linear equation is an algebraic equation where each term has an exponent of 1 and when this equation is graphed, it always results in a straight line.
Given,
Company A charges an initial fee of $15 for the rental plus $1 per mile driven
f(x) = 15 + x
Company B charges an initial fee of $45 for the rental plus $0.50 per mile driven
g(x) = 45 + 0.5x
g(x) = 45 + x/2
x is number of miles
For 80 miles company A charges
f(80) = 15 + 80
= 95
For 80 miles company B charges
g(80) = 45 + 80/2
= 85
By the above calculation, Company B is cheaper than Company A for 80 miles drive.
Hence,
f(x) = 15 + x is Equation for the cost of company A
g(x) = 45 + x/2 is Equation for the cost of company B
Company B is cheaper than company A for drive of 80 miles.
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Write True if the statement is correct and Faise if the statement is incorrect
1. Every composite number can be expressed as a product of primes
, 2. Some quadratic equation has infinite solution
3. For any two real numbers x andythenix-yl=ly=xl
4. 2(3+√7)-2√7 = 6√7
5. 25,471,287 is not divisible by 3 and 9,
6. √6 is not rational number,
7. Between every two real numbers there is another real number,
8. For any natural numbers a and b GCF (a, b) * LCM(a,b) = a*b.
9. Integer is not closed under division,
10. The product of non zero rational number and irrational number is rational⁹
1. Every composite number can be expressed as a product of primes - True
2. Some quadratic equation has infinite solution - True
3. False (ix-yl = |x-y|, which may not equal ly = |y|)
What is Rational Number?A rational number is a number that can be expressed as the ratio of two integers.A rational number can be written in the form p/q, where p and q are integers and q ≠ 0.Examples of rational numbers include -3/4, 0, 1/2, and 5.Every integer is a rational number, since it can be expressed as the ratio of the integer and 1.Every fraction can be represented as a rational number, since it is the ratio of two integers.The decimal representation of a rational number will either terminate (e.g. 1/4 = 0.25) or repeat (e.g. 1/3 = 0.3333...).The set of rational numbers is dense, meaning that between any two rational numbers, there is another rational number.4. 2(3+√7)-2√7 = 6√7 - False
5. False (25,471,287 is divisible by 3)
6. √6 is not rational number - True
7. Between every two real numbers there is another real number -True
8. For any natural numbers a and b GCF (a, b) * LCM (a, b) = a*b. -True
9. Integer is not closed under division -True
10. False (The product of a non-zero rational number and an irrational number is not necessarily rational)
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Compute the probability of x successes in the n independent trials of the experiment. n=9, p=0.5, x≤3
The probability of x successes in the n independent trials of the experiment is 0.1640625
What is probability?Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.
Given that, n=9, p=0.5, x≤3 we need to find probability of x successes in the n independent trials of the experiment.
We know that,
P(X=x) = ⁿCₓpˣq⁽ⁿ⁻ˣ⁾
q = 1-p = 1-0.5
q = 0.5
P(X=x) = ⁹C₃(0.5)³(0.5)⁹⁻³
P(X=3) = ⁹C₃ (0.5)³(0.5)⁶
⁹C₃ = 9! / 3!(9-3)! = 9!/3! × 6!
⁹C₃ = 84
P(X=3) = 84×0.125×0.015625
= 0.1640625
Hence, the probability of x successes in the n independent trials of the experiment is 0.1640625
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