The range of a data set is the difference between the highest and lowest numbers in the set. To find the range for the given set of data, we first need to find the highest and lowest values in the set. From the given set of data, we can see that the minimum value is 3 and the maximum value is 12. Therefore, the range of the given set of data is 12 - 3 = 9.
Hence, the answer is 9.
MODELING REAL LIFE The volume of the largest of the six pyramids constructed by the Norte Chico people in Caral, Peru, is about 4,500,000 cubic feet. What is the height of the pyramid? A pyramid with the height h feet. Length of 8 h minus 30 feet. Width of 8 h plus 20 feet. Height: ft
The height of the pyramid whose volume is already given would be = 1,440,000 ft
How to calculate the height of the given pyramid?To calculate the height of the pyramid, the formula of the Volume of the pyramid should be used.
Volume = 1/3lwh
where l = base length = 8h-30 ft= 3.75 ft
w = base width= 8h+20 ft = -2.5 ft
h = h ft
The volume of the pyramid = 4,500,000 cubic feet
4,500,000 = 1/3 × (3.75)(-2.5)h
Make h the subject of formula;
h = 4500000×3/3.75×(-2.5)
h = 13500000/9.375
h = 1,440,000 ft.
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How to perform bootstrapped likelihood ratio test?
To perform a bootstrapped likelihood ratio test, you need to fit the two nested models to your data, calculate the likelihood ratio test statistic, generate bootstrap samples, fit the two models to each bootstrap sample, calculate the p-value, and interpret the results.
The bootstrapped likelihood ratio test is a statistical method used to compare the goodness-of-fit of two nested models. Here's how you can perform a bootstrapped likelihood ratio test
Fit the two nested models: Start by fitting the two nested models to your data. One model is a restricted model, and the other is an unrestricted model that contains additional parameters.
Calculate the likelihood ratio test statistic: Calculate the likelihood ratio test statistic by subtracting the log-likelihood of the restricted model from the log-likelihood of the unrestricted model. This gives you the test statistic (often denoted as "LR").
Generate bootstrap samples: Generate a large number of bootstrap samples from the original dataset. Each bootstrap sample should be the same size as the original dataset and drawn with replacement.
Fit the two models to each bootstrap sample: For each bootstrap sample, fit the two nested models and calculate the likelihood ratio test statistic as described in steps 1 and 2.
Calculate the p-value: Calculate the p-value as the proportion of bootstrap samples for which the likelihood ratio test statistic is greater than or equal to the test statistic calculated from the original dataset.
Interpret the results: If the p-value is less than your chosen significance level (e.g., 0.05), you can reject the null hypothesis that the restricted model is better than the unrestricted model in favor of the alternative hypothesis that the unrestricted model is better.
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Question 4 Part A (2 points): Landon has $16 and wants to buy a combination of sandwiches and chips to feed his 4 teammates. Each sandwich costs $4 and each bag of chips costs $2. This system of inequalities models the scenario.
4x + 2y ≤ 16
x + y ≥ 4
Describe the graph of the system.
A.
A graph of two solid lines that intersect at point (0, 4).
B.
A graph of two solid lines that intersect at point (4, 0).
C.
A graph of one solid line and one dashed line that intersect at point (0, 4).
D.
A graph of one solid line and one dashed line that intersect at point (4, 0).
The correct statement about the graph is,
A graph of two solid lines that intersect at point (4, 0).
Given that;
This system of inequalities models the scenario.
4x + 2y ≤ 16
And, x + y ≥ 4
Now, We can draw the inequalities in graph, we get;
The intersection point of the inequality is, (4, 0)
Thus, The correct statement about the graph is,
A graph of two solid lines that intersect at point (4, 0).
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If s(d) represents the number of songs downloaded in a year d, what is the interpretation of s(2020) = 5,220,000.
A. 5,220,000 songs were downloaded in the year 2020.
B. There is not enough information to interpret the information.
C. In the year 2020 $5,220,000 was earned from downloaded songs.
D. 2,020 songs were downloaded at a cost of $5,220,000.
Answer: A. 5,220,000 songs were downloaded in the year 2020.
Step-by-step explanation: The interpretation of s(2020) = 5,220,000 is that in the year 2020, 5,220,000 songs were downloaded. The function s(d) represents the number of songs downloaded in a year d, so plugging in 2020 for d gives us the specific number of songs downloaded in that year.
the fruity tooty company claims that the population proportion for each of its five flavors is exactly 20%. jane counts 92 red candies in a 400-count sample.z equals fraction numerator p with hat on top minus p over denominator square root of begin display style fraction numerator p q over denominator n end fraction end style end root end fraction using the formula and data provided, what is the value of the z-test statistic? answer choices are rounded to the hundredths place.
Value of the z-test statistic is 1.20.
How to find the value of the z-test statistic?To find the value of the z-test statistic, we need to calculate the sample proportion and the standard error of the proportion.
The sample proportion is:
p⁻ = 92/400 = 0.23
The standard error of the proportion is:
SE = √[(pq)/n] = √[(0.20.8)/400] = 0.025
where p = 0.2 and q = 0.8.
Now we can calculate the z-test statistic:
z = (p⁻ - p)/SE = (0.23 - 0.2)/0.025 = 1.2
Therefore, the value of the z-test statistic is 1.20.
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the cylinder with the height of 4 M has a volume of 2,827.43 cubic meters find the length of the diameter
[tex]\textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ V=2827.43\\ h=4 \end{cases}\implies 2827.43=\pi r^2(4)\implies \cfrac{2827.43}{4\pi }=r^2 \\\\\\ \sqrt{\cfrac{2827.43}{4\pi }}=r\hspace{5em}\stackrel{\textit{twice that is the \underline{diameter}}}{2\sqrt{\cfrac{2827.43}{4\pi }}} ~~ \approx ~~ \text{\LARGE 30}[/tex]
Mr. Frost worked 37 hours last week. He was paid $17 per hour. How much money did he make last week?
Answer:
$481
Step-by-step explanation:
37×$17=$481
he made $481
A ( x + 3 ) < 5x + 15 - x
find the value for a for which the solution of the inequality is all real numbers
The value of a for which the solution of the inequality is all real numbers is a = 5.
We have to first simplify the inequality:
A(x + 3) < 5x + 15 - x
Ax + 3A < 4x + 15
Ax - 4x < 15 - 3A
Simplifying further:
x(A - 4) < 15 - 3A
Now, we need the inequality to hold for all real values of x. This means that the coefficient of x, (A - 4), must have a fixed sign, and that the right-hand side, 15 - 3A, must be unbounded.
For the coefficient of x to have a fixed sign, we need either A - 4 < 0 or A - 4 > 0. This means that A must be less than 4 or greater than 4.
For the right-hand side to be unbounded, we need 15 - 3A to be equal to positive or negative infinity. Since 15 - 3A is a linear function of A, this only occurs when A = 5.
So, the solution to the inequality for all real numbers is:
if A < 4, then the solution is x < (15 - 3A) / (4 - A)
if A > 5, then the solution is x > (15 - 3A) / (4 - A)
if A = 5, then the solution is all real numbers.
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Find the length of each segment.
13. YW and WZ
The length of each segment is:
YW = 5 units
WZ = 8 units
How to find the length of each segment?The length of each segment can be found by using "bisector of an angle in a triangle theorem".
The theorem states that "the bisector of an angle in a triangle theorem divides the opposite sides in the ratio of the sides containing the angle".
Applying the theorem on the given triangle, we can say:
XY/XZ = YW/WZ
8/12.8 = (t/2)/(t-2)
8(t-2) = 12.8(t/2)
8t - 16 = 6.4t
8t - 6.4t = 16
1.6t = 16
t = 10
YW = t/2
YW = 10/2
YW = 5 units
WZ = t - 2
WZ = 10 - 2
WZ = 8 units
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you are constructing a box from a piece of cardboard with dimensions 5 by 7 meters. you cut equal-size squares from each corner of the cardboard so you may fold the edges to construct the open top box. what are the dimensions of the box with the largest volume?
3.1634 m , w = 1.1634 , h = 1.91834 m are the dimensions of the box with the largest volume.
What is a simple volume definition?
Volume is defined as the space occupied within the bounds of an object in three-dimensional space. Also called the capacity of the object. The basic formula for the area of a rectangle is length x width, but the basic formula for volume is length x width x height.
Length of cardboard = 7 m
width of a cardboard = 5 m
let x be the side of square of piece cut from each corner
length of box = (7 -2x) m
width of box = (5 - 2x) m
height of box = x
volume V = (7 -2x) * (5 - 2x) * X
= (35 - 14X - 10X + 4X²) * X
= 4X³ - 24X² + 35X
If v is max v' = 0 and v'' < 0
v' = 12x² - 48x + 35
x = 6.0816 or 1.91834
if x = 6.0816 , box will not be follow.
x = 1.91834
v''(x) = 24x - 48
= 46.04 - 48 = -1.95 < 0
V is max.
length = 7 - 3.8366 = 3.1634 m
width = 5 - 3.8366 = 1.1634 m
height = 1.91834 m
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a report claims that the proportion of all adults who are vegetarians is 0.12. believing this claimed value is incorrect, a researcher surveys a large random sample of 1,000 adults and finds that 140 of the adults in the sample are vegetarians. what will the test statistic be equal to in this example? as you are engaging in calculations, try not to do a lot of rounding until you get to the very end, and choose the answer below that is closest to what you calculate. a. 2.5 b. 1.4 c. 1.9 d. 0.3 d. less than 0.1
The correct answer is (c) 1.9
How to calculate the test statistic?To calculate the test statistic in this example, we can use the formula for a z-test for proportions:
[tex]z = (p - P) / \sqrt{[P(1-P) / n][/tex]
where:
p is the sample proportion of vegetarians (140/1000 = 0.14)
P is the hypothesized population proportion of vegetarians (0.12)
n is the sample size (1000)
Substituting these values into the formula, we get:
[tex]z = (0.14 - 0.12) / \sqrt{[0.12(1-0.12) / 1000][/tex]
z = 0.02 / 0.0110
z = 1.818 (rounded to three decimal places)
Therefore, the test statistic in this example will be approximately equal to 1.818. The answer closest to this value is (c) 1.9
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select the correct answer. in a box containing 20 cans, 12 of the cans contain vegetables. in the box, 5 of the cans are dented, with 2 of the dented cans containing vegetables. what is the probability that a randomly selected can in the box is dented or contains vegetables?
The probability that a randomly selected can in the box is dented or contains vegetables is 19/20.
To solve the problem, we need to use the formula for the probability of the union of two events:
P(A or B) = P(A) + P(B) - P(A and B)
where A and B are two events.
In this case, we want to find the probability that a randomly selected can is dented or contains vegetables. Let's call these events D and V, respectively.
We are given that there are 20 cans in total, 12 of which contain vegetables, and 5 of which are dented, with 2 of the dented cans containing vegetables. This information allows us to calculate the probabilities of the individual events
P(D) = 5/20 = 1/4
P(V) = 12/20 = 3/5
P(D and V) = 2/20 = 1/10
To find the probability of the union of these events, we plug these values into the formula
P(D or V) = P(D) + P(V) - P(D and V)
= 1/4 + 3/5 - 1/10
= 19/20
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If line segment AC = 9 and Angle A = 34°, find line segment BC. Round to the nearest tenth.
Answer: BC ≈ 6.4 units
Step-by-step explanation: We can use trigonometry to find the length of line segment BC.
Since we know the length of AC and the measure of angle A, we can use the trigonometric function cosine (cos) to calculate BC.
Using the formula: cos(A) = Adjacent / Hypotenuse, we have
cos(34°) = BC / 9.
Solving for BC, we get BC ≈ 6.4 units (rounded to the nearest tenth).
Here is an article on trigonometry basics: https://byjus.com/maths/trigonometry/#:~:text=Trigonometry%20Basics,are%20based%20on%20these%20functions.
The dolphins at the San Diego zoo eat 3 1/8 buckets of fish each day. The killer whales eat 2 3/5 times as much fish as the dolphins
how many buckets of fish do the killer whales eat each day?
PLEASE HELP ILL MARK U AS BRAINLIEST!!
Answer: 94 square inches 2
Step-by-step explanation:
2×(5×4 + 5×3 + 4×3) = 94 sq in 2
length x width x height
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how many intervals (or 'bins' or 'classes') should be chosen when creating a histogram? question 1 options: most often, about 8-10. eleven. it can vary - it really depends on the distribution of the variable. a minimum of 5.
"It can vary - it really depends on the distribution of the variable."
The number of intervals, or bins, to choose when creating a histogram can vary depending on the distribution of the variable.
Most often, about 8-10 intervals are used, but there is no set rule. It is generally recommended to have at least 5 intervals, but if the data is highly skewed or has outliers, more intervals may be needed to accurately represent the distribution.
Ultimately, the goal is to choose a number of intervals that provides a clear visual representation of the data without oversimplifying or overcomplicating the histogram.
The number of intervals or bins to be chosen when creating a histogram can vary and it really depends on the distribution of the variable.
While most often, about 8-10 bins are used, there is no hard and fast rule for the number of bins to be used in a histogram.
In general, the number of bins should be large enough to display the shape of the distribution clearly, but not so large that it obscures important features of the distribution or leads to overfitting.
A minimum of 5 bins is recommended to display the basic shape of the distribution, but more bins may be necessary for complex or multi-modal distributions.
Depending on the distribution of the variable, a histogram's number of intervals or bins can be altered.
There is no established guideline, however 8–10 intervals are typically utilized.
A minimum of five intervals are often advised, however if the data is extremely skewed or contains outliers, more intervals could be required to correctly depict the distribution.
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Tell whether a triangle can have the given angle measures. 115. 1∘, 47. 5∘, 93∘
No, a triangle cannot have the given angle measures, 115. 1∘, 47. 5∘, 93∘, as sum of these is greater than 180°.
The angle sum of a triangle will always be equal to 180°. The angle sum of a quadrilateral is equal to 360°, and a triangle can be created by slicing a quadrilateral in half from corner to corner. Since a triangle is essentially half of a quadrilateral, its angle measures should be half as well.
The sum of the angles in a triangle is always 180°, and 115.1° + 47.5° + 93° = 255.6°, which is greater than 180°.
Therefore, it is not possible for a triangle to have these angle measures.
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011.considering grade c or above as a pass grade, how many students from this data successfully passed the course?
If we consider grade "C" or above as a pass grade, then the number of students who successfully passed the course are 10 students.
The Number of students who got grade "A" = 8,
The Number of students who got "B" grade is 2,
The Number of students who got grade "C" = 0,
The Number of students who got grade "D" = 0,
The Number of students who got grade "F" = 1,
Since the question asks for the number of students who successfully passed the course (i.e., grade "C" or above is considered a pass grade),
We add the number of students who got grades "A" and "B".
So, total number of students who successfully passed course are :
⇒ 8 (students who got grade "A") + 2 (students who got grade "B")
⇒ 10
Therefore, 10 students successfully passed the course.
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The given question is incomplete, the complete question is
If 8 students of the class got grade "A", 2 student got grade "B", No student got grade "C" and "D", 1 student got grade "F".
Considering grade "C" or above as a pass grade, how many students from this data successfully passed the course?
Round to the nearest 10th a cylinder is 22 inches and 12.5 inches what is the surface area 
Which number line represents the solution to the inequality
Answer:
B
Step-by-step explanation:
we randomly select 100 pell grant recipients from two states. state a is a relatively small state with approximately 4,000 pell grant recipients. state b is a large state with approximately 200,000 pell grant recipients. suppose that the mean and standard deviation in individual pell grants is approximately the same for both states: and . for which state is the sample mean for our 100 pell grant recipients most likely to be within $80 of $2,600?
The sample mean for 100 pell grant recipients is more likely to be within $80 of $2,600 for State A, given the relatively smaller population size and the use of the t-distribution.
To compare the two states, we need to calculate the probability of the sample mean for 100 pell grant recipients being within $80 of $2,600 for each state.
For State A, with only 4,000 pell grant recipients, the sample size of 100 is relatively large, but not enough to use the normal distribution. Therefore, we need to use the t-distribution, which has fatter tails than the normal distribution, making it more likely to produce extreme values. This means that the probability of the sample mean for 100 pell grant recipients being within $80 of $2,600 is higher for State A than for State B.
For State B, with a larger population of 200,000 pell grant recipients, the sample size of 100 is a small fraction of the population, allowing us to use the normal distribution. However, as the distribution is narrower than the t-distribution, the probability of the sample mean for 100 pell grant recipients being within $80 of $2,600 is lower for State B than for State A.
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an lti system has the frequency response function h(ω)=1/(jω 3). compute the output if the input is
The output of the LTI system with frequency response function [tex]h(w) = 1/(jw^3)[/tex], given an input[tex]x(t)[/tex], is:
[tex]y(t)[/tex]= inverse Fourier transform of [tex][1/(jw^3) X(w)][/tex]
To compute the output of an LTI system with frequency response function [tex]h(w) = 1/(jw^3)[/tex], given an input x(t), we can use the Fourier transform:
[tex]H(w)[/tex] = Fourier transform of [tex]h(t)[/tex]
[tex]X(w)[/tex] = Fourier transform of [tex]x(t)[/tex]
[tex]Y(w) = H(w) X(w)[/tex]
[tex]Y(w)[/tex] is the Fourier transform of the output [tex]y(t)[/tex].
Using the given frequency response function, we have:
[tex]H(w) = 1/(jw^3)[/tex]
Taking the Fourier transform of the input [tex]x(t)[/tex], we have:
[tex]X(w)[/tex] = Fourier transform of [tex]x(t)[/tex]
And multiplying [tex]H(w)[/tex] and[tex]X(w)[/tex], we get:
[tex]Y(w) = H(w) X(w) = 1/(jw^3) X(w)[/tex]
Taking the inverse Fourier transform of [tex]Y(w)[/tex], we get the output [tex]y(t)[/tex]:
[tex]y(t)[/tex] = inverse Fourier transform of [tex]Y(w)[/tex]
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A 150 pound crate is used to hold the boxes on the forklift. What is the maximum number of boxes that the forklift can carry in the crate.
Answer: To determine the maximum number of boxes that the forklift can carry in the crate, we need to know the weight of each box and the weight capacity of the forklift. Let's assume that each box weighs 20 pounds and the weight capacity of the forklift is 2000 pounds.
To find the maximum number of boxes, we need to subtract the weight of the crate from the weight capacity of the forklift and then divide by the weight of each box:
Maximum number of boxes = (Weight capacity of forklift - Weight of crate) / Weight of each box
Maximum number of boxes = (2000 - 150) / 20
Maximum number of boxes = 1850 / 20
Maximum number of boxes = 92.5
Since we cannot have a fraction of a box, we need to round down to the nearest whole number. Therefore, the forklift can carry a maximum of 92 boxes in the crate.
Step-by-step explanation:
Andy has 3.2 x 10³ dollars in his bank account. His brother Dan has 4.7 times that amount in his account. How much does Dan have in his account? Write the amount in scientific notation.
Answer:
1.504 x 10^4
Step-by-step explanation:
you want to obtain a sample to estimate a population mean. based on previous evidence, you believe the population standard deviation is approximately . you would like to be 90% confident that your estimate is within 2.5 of the true population mean. how large of a sample size is required? n
The required sample size of the given estimate is 1090.9809
The population standard deviation is approximately σ = 50.2. You would like to be 90% confident that your estimate is within 1.5 of the true population mean.
What is Statistic?Statistics is the study of mathematics that deals with relations between comprehensive data.
Here,
For the 90% confidence the value of the z = 1.645
Now the sample size is given as,
n = [z × σ/2.5]²
Substitute values in the above equation,
n = [1.645×50.2/2.5]²
n = 1090.9809
Thus, the required sample size of the given estimate is 1090.9809.
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the graph below displays the amount of time to the nearest hour spent on homework per week for a sample of students. which measures of center and variability would be most appropriate to describe the given distribution?
The measures of center and variability that would be most appropriate to describe the given distribution is D) Median and IQR.
What is median and IQR?The middle value in a set of data is represented by the median, a measure of central tendency. It is the value that, when a dataset's values are ranked from lowest to highest, distinguishes the lower from the upper half of the dataset.
The middle 50% of a dataset's interquartile range (IQR) is a measure of variability that captures this dispersion. It is the difference between the data's first (Q1) and third (Q3) quartiles. The number that divides the lowest 25% of the data from the remaining data is known as the first quartile, and the value that separates the highest 25% of the data from the remaining data is known as the third quartile.
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The complete question is:
Dylan bought 3 identical shirts online for a total cost of $71.83 including a flat rate of $7.99 for shipping. Complete an equation to find the cost of each shirt, s, using the numbers below.
Answer:
$21.28
Step-by-step explanation:
We Know
Dylan bought 3 identical shirts online for a total cost of $71.83
$7.99 for shipping
We have the equation:
71.83 = 3s + 7.99
63.84 = 3s
s = $21.28
So, each shirt cost $21.28
the rear window wiper blade on a car has a length of 10 inches. the blade is mounted on a 13 inch arm, 3 inches from the pivot point. if the wiper turns through an angle of 115 degrees, how much area is swept clean?
The area swept clean by the rear window wiper blade is approximately 91.37 square inches.
We'll calculate the area swept clean by the rear window wiper blade using the given information.
Convert the angle from degrees to radians:
115 degrees * (π radians / 180 degrees) = (23π/36) radians.
Find the length of the arc that the wiper blade traces out:
Arc length = (radius * angle) = (10 inches * (23π/36) radians) ≈ 20.08 inches.
Calculate the area of the sector formed by the arc and the pivot point:
Sector area = (1/2) * (radius² * angle) = (1/2) * (10 inches² * (23π/36) radians) ≈ 100.42 square inches.
Find the length of the arc that the mounting point of the blade traces out:
Arc length = (radius * angle) = (3 inches * (23π/36) radians) ≈ 6.02 inches.
Calculate the area of the sector formed by this arc and the pivot point:
Sector area = (1/2) * (radius² * angle) = (1/2) * (3 inches² * (23π/36) radians) ≈ 9.05 square inches.
Subtract the smaller sector area from the larger one to get the area swept clean by the wiper blade:
Area swept clean = 100.42 square inches - 9.05 square inches ≈ 91.37 square inches.
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The area swept clean by the rear window wiper blade is approximately 31.53 square inches.
To determine the area swept clean by the rear window wiper blade, we first need to find the length of the arc covered by the blade. We can use the formula:
arc length = radius * angle in radians
To convert the angle from degrees to radians, we multiply by[tex]\pi /180[/tex]:
115 degrees x [tex]\pi /180[/tex] = 2.007 radians
The radius is the length of the wiper arm, which is 13 inches minus the distance from the pivot point to the blade, which is 3 inches. Therefore, the radius is:
13 inches - 3 inches = 10 inches
Now we can calculate the arc length:
arc length = 10 inches x 2.007 radians = 20.07 inches
So the area swept clean by the wiper blade is approximately equal to the area of the sector formed by the arc and the two radii. The formula for the area of a sector is:
area = (angle in radians / [tex]2\pi[/tex]) x [tex]\pi r^2[/tex]
Plugging in values we have:
area = (2.007 / [tex]2\pi[/tex]) x [tex]\pi[/tex][tex](10 inches)^2[/tex] = 31.53 square inches
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#14 Please Help!!!!!!!!
The measure of the angle m∠FGH is 83°.
How to find the measure of m∠FGH?
Since the measure of an angle formed by a two tangents drawn from a point outside the circle is one-half the the difference of the intercepted arcs.
Thus, we can say:
m∠FGH = 1/2 * (∠FIH - 97°)
∠FIH = 360 - 97 = 263° (sum of angle in a circle)
m∠FGH = 1/2 * (263 - 97°)
m∠FGH = 1/2 * 166
m∠FGH = 83°
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Point A = (5,4). If you rotated A 90 degrees about the point (2,-1), what would be the coordinates of A'?
Point A = (5,4). If you rotated A 90 degrees about the point (2,-1), the coordinates of A' is (-1,2).
Describe Rotation?Rotation is the process of rotating an object or a point around a fixed point or axis. In mathematics, rotation refers to a transformation that preserves the size and shape of an object while changing its orientation. It is a basic geometric transformation that is used in various fields, including mathematics, physics, engineering, and computer graphics.
In a two-dimensional space, a rotation is typically described by an angle of rotation and a fixed point, which is known as the center of rotation. The angle of rotation represents the amount by which the object is rotated, while the center of rotation is the point around which the object is rotated. In a three-dimensional space, a rotation is described by an axis of rotation and an angle of rotation.
To rotate point A 90 degrees counterclockwise about the point (2,-1), we can use the following formula:
A' = (x', y') = (a + (x - a) cosθ - (y - b) sinθ, b + (x - a) sinθ + (y - b) cosθ)
where (a,b) is the center of rotation and θ is the angle of rotation (90 degrees in this case).
Substituting the given values, we get:
a = 2, b = -1, x = 5, y = 4, θ = 90 degrees
x' = 2 + (5 - 2) cos(90) - (4 + 1) sin(90) = 2 - 3 = -1
y' = -1 + (5 - 2) sin(90) + (4 + 1) cos(90) = -1 + 3 = 2
Therefore, the coordinates of A' are (-1, 2).
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