What is the x-intercept for the following standard form line?
−3x+5y=72

Answer:

$10.40

Step-by-step explanation:

If each packet of hamburger buns contains 8 buns, and you need 64. You do 64/8 = 8

So you need 8 packets of buns

And $1.30 x 8 = $10.40

If the target behavior is recorded whenever it occurs at any point within the scoring interval, the measurement procedure used is called continuous recording or event recording. This method involves observing and recording each instance of the behavior without considering its duration or intensity.

Continuous recording is typically used when the target behavior is discrete and occurs relatively infrequently or has a short duration. With this measurement procedure, each occurrence of the behavior is recorded as a separate event, regardless of when it starts or ends within the scoring interval. Examples of behaviors that can be measured using continuous recording include vocalizations, hand clapping, or instances of aggression.

Continuous recording provides a comprehensive account of the frequency or rate at which the behavior occurs, allowing for a detailed analysis of its occurrence patterns. This method is particularly useful when a precise count of behavior instances is needed and when the duration or intensity of the behavior is not a focus of measurement.

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The effects of the reduction in each statistical measure of the data-set is given as follows:

a. The shape of the data-set remains constant.

b. Both the mean and the median will be subtracted by 500.

c. The standard deviation and the IQR remain constant.

The shape of the distribution is related to how the distributions are spread around the mean.

Each observation will be reduced by $500, meaning that the spreads remain constant, hence:

The shape of the data-set remains constant.

The mean is the sum of the observations divided by the number of observations.

500 will be subtracted by each observation, hence the mean will be subtracted by 500.

The median is the middle value of the data-set. Since all values will be subtracted by 500, the middle value also will, hence the median will be subtracted by 500.

Both are measures of spread, as:

Since all measures change by the same amount, the difference between the percentiles will be the same, as will the differences squared between each observation and the mean, hence:

The standard deviation and the IQR remain constant.

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On a unit circle, the terminal point of theta is (sqrt 3/2, 1/2). theta if 0° is 30 degrees.

The measurement of an angle is expressed in terms of degrees. Radians and degrees are the two most often used units for measuring angles. In actual geometry, the angle is always measured in degrees. Degree is denoted by the sign °. (degree symbol). One full rotation is equal to 360 degrees (sometimes written as 360°), which is the measurement of a complete angle in degrees. The unit of measurement for angles is the degree. Since radians are the SI unit for measuring angles, it is not a SI unit. In geometry, we typically use a protractor to measure angles in degrees. In most cases, a protractor is utilized in schools to measure angles in order to address various mathematical issues.

Given

x = √3/2

y = 1/2

theta = arctan(y/x)

theta = arctan(1/2÷√3/2)

theta = arctan(1/2 * 2/√3)

theta = arctan(1/√3)

theta = 30°

On a unit circle, the terminal point of theta is (sqrt 3/2, 1/2). theta if 0° is 30 degrees.

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Answer:

Step-by-step explanation:

Answer:

(C)DC = 6 StartRoot 3 EndRoot in.

(D)AC = 12 StartRoot 3 EndRoot in.

Step-by-step explanation:

Step-by-step explanation:

if you could solve 3a and 3b, you can also solve 3c. it is the same principle.

x-intercepts means y = 0.

remember, a square root has always 2 results : a positive and a negative one.

3c

0 = -(x - 3)² + 25

(x - 3)² = 25

x - 3 = ±5

x1 = +5 + 3 = 8

x2 = -5 + 3 = -2

3a

0 = (x + 1)² - 4

4 = (x + 1)²

±2 = x + 1

x1 = +2 - 1 = 1

x2 = -2 - 1 = -3

3b

0 = (x + 2)² - 9

9 = (x + 2)²

±3 = x + 2

x1 = +3 - 2 = 1

x2 = -3 - 2 = -5

The number of vegetable bearing plants are : 0.45x, where x is the total number of plants.

A fraction in mathematics represents a portion or element of the whole. It represents the appropriate parts of the totality. A fraction is made up of two parts: the numerator and the denominator. The number at the top is the numerator, and the number at the bottom is the denominator. The numerator indicates the number of equal parts that were actually taken, whereas the denominator shows the total number of equal parts in the whole.

Let x be the initial plant population.

Now that 1/10 of plants bear fruit, the quantity of fruit-bearing plants is equal to 0.1x.

Plants remaining then equal x - 0.1x = 0.9x.

Once more, because half of the remaining plants give flowers, the number of flower-bearing plants is equal to 0.5(0.9x) = 0.45x.

As a result, the number of plants left is equal to 0.9x - 0.45x = 0.45x.

Given that all remaining plants produce veggies, the number of plants that do so is equal to 0.45.

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If a certain examination is to be taken by five students independently of one another, and the number of minutes required by any particular student to complete the examination has the exponential distribution for which the mean is 80 and the examination begins at 9:00 a.m, then the probability that at least one of the students will complete the examination before 9:40 a.m is 80.6%.

To find the probability that at least one of the students will complete the examination before 9:40 a.m, follow these steps:

Therefore, the probability that at least one of the five students will complete the examination before 9:40 a.m is 0.8060 or 80.6%.

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Answer:

The least possible height of the flagpole is 227.33 inches and the greatest possible height is 248 inches

Step-by-step explanation:

Given

See attachment for the system

Required

Determine the least and greatest heights of the flagpole

From the question, we have:

\(AB = 7\ to\ 7\frac{1}{2} ft\)

\(BE =62in\)

\(BC = 20\ to\ 21ft\)

Convert AB and BC to feet (multiply by 12)

\(AB = 84in\ to\ 90in\)

\(BC = 240in\ to\ 252in\)

The height of the flagpole is represented as: CD

To calculate CD, we make use of the following equivalent ratios:

\(CD : AC = BE : AB\)

Express as fraction

\(\frac{CD }{ AC } = \frac{BE }{ AB}\)

Make CD the subject

\(CD = \frac{BE }{ AB} * AC\)

From the attachment:

\(AC = AB + BC\)

Where:

\(AB = 84in\ to\ 90in\)

\(BC = 240in\ to\ 252in\)

The possible values of AC are:

\(AC = 84 + 240 =324in\)

\(AC = 84 + 252 =336in\)

\(BC = 90 + 240 = 330in\)

\(AC = 90 + 252 =342in\)

So:

\(CD = \frac{BE }{ AB} * AC\)

\(BE =62in\)

When AC = 324 ; AB = 84

\(CD = \frac{62}{84} * 324\)

\(CD = \frac{62* 324}{84}\)

\(CD = \frac{20088}{84}\)

\(CD = 239.14in\)

When AC = 336 ; AB = 84

\(CD = \frac{62}{84} * 336\)

\(CD = \frac{62* 336}{84}\)

\(CD = \frac{20832}{84}\)

\(CD = 248in\)

When AC = 330 ; AB = 90

\(CD = \frac{62}{90} * 330\)

\(CD = \frac{62 * 330}{90}\)

\(CD = \frac{20460}{90}\)

\(CD = 227.33in\)

When AC = 342 ; AB = 90

\(CD = \frac{62}{90} * 342\)

\(CD = \frac{62* 342}{90}\)

\(CD = \frac{21204}{90}\)

\(CD = 235.6in\)

So, we have:

\(CD = 239.14in\)

\(CD = 248in\)

\(CD = 227.33in\)

\(CD = 235.6in\)

The least possible height of the flagpole is 227.33 inches and the greatest possible height is 248 inches

The given range of the length of the shadow and distance of the student

from the tree of between 7 and \(7\frac{1}{2}\) feet and 21 and 22 feet gives;

The greatest possible height of the flagpole in feet ≈ 21.8 feet

The least possible height of the flagpole in feet = 19.3 feet

Height of the student = 62 inches = \(5\frac{1}{6}\) feet

Distance of the flagpole to the student = 21 to 22 feet = 252 to 264 inches

Length of the student's shadow = Between 7 and \(\mathbf{7\frac{1}{2}}\) feet = Between 84 and 90 inches

Required:

The least and greatest height of the flagpole.

Solution;

By using similar triangles formed by the student and the flagpole, we

have;

\(\dfrac{Height \ of \ the \ student}{Length\ of \ the \ student's \ shadow} = \mathbf{\dfrac{Height \ of \ the \ flagpole}{Length\ of \ the \ flagpole's \ shadow}}\)

Which gives;

\(\dfrac{62}{84 \ or \ 90} =\mathbf{ \dfrac{h}{252+(84 \ or \ 90) \ or \ 264 + (84 \ or \ 90)}}\)

Based on the method of cross multiplication, h will be the greatest

possible value, when we have, the largest possible value for the

denominator on the right hand side and the least possible value for the

denominator on the left hand side of the equation, and vice versa.

Which gives;

\(\dfrac{62}{90} = \mathbf{\dfrac{h_{min}}{252 + 84}}\)

\(h_{min} \ in \ feet = \mathbf{\dfrac{\dfrac{62}{90} \times (252 + 84)}{12}} \approx 19.3\)

The least possible height of the flagpole in feet, \(h_{min}\) ≈ 19.3 feet

Similarly. we have;

\(\dfrac{62}{84} = \mathbf{\dfrac{h_{max}}{264 + 90}}\)

Which gives;

\(h_{max} = \dfrac{62}{84} \times (264 + 90) \approx 261.3\)

\(h_{max} \ in \ feet \approx \dfrac{261.3}{12} \approx\mathbf{ 21.8}\)

The greatest possible height of the flagpole in feet, \(h_{max}\) ≈ 21.8 feet

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√27 is greater than √95/2. So the sign 'Greater than' will be used to satisfy this statement: √27  > √95/2

In the given question, we have to put the appropriate sign of inequality so that statement becomes True. First, we will have to calculate the roots of both √27 and √95/2.

Calculating,  => √27  =  5.19

and,  => √95/2 = 4.87

Comparing the square roots of both √27 and √95/2 we can clearly see that √27 is greater than √95/2. So, we get √27 > √95/2                    

Hence, the sign used to make the statement true will be of 'Greater than' such that √27 > √95/2

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there are 4,200 possible distributions of the 10 different types of cookies among Anthony, Misha, and Lizard, considering the given conditions.

To determine the number of possible distributions of the 10 different types of cookies among Anthony, Misha, and Lizard, we can use combinatorial analysis.

First, let's consider the distribution of 3 cookies to Anthony. Since there are 10 cookies to choose from, the number of ways to select 3 cookies for Anthony is given by the combination formula:

C(10, 3) = 10! / (3! * (10 - 3)!) = 10! / (3! * 7!) = 10 * 9 * 8 / (3 * 2 * 1) = 120

Next, we consider the distribution of 3 cookies to Misha. Since there are 7 remaining cookies after Anthony's selection, the number of ways to select 3 cookies for Misha is:

C(7, 3) = 7! / (3! * (7 - 3)!) = 7! / (3! * 4!) = 7 * 6 * 5 / (3 * 2 * 1) = 35

Finally, the remaining 4 cookies are automatically assigned to Lizard.

To find the total number of possible distributions, we multiply the number of ways to distribute the cookies to Anthony, Misha, and Lizard:

Total number of possible distributions = 120 * 35 = 4,200

Therefore, there are 4,200 possible distributions of the 10 different types of cookies among Anthony, Misha, and Lizard, considering the given conditions.

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The area of the Temasa's doll house is 31.5 square feet.

What is the scale factor?

The scale factor is a ratio that compares the measurements of an object on a drawing or map to the actual measurements of that same object in real life. It is usually expressed as a ratio in the format 1:n (or 1/n) where n is the scale factor.

We can start by using the scale factor of 1:10 to find the actual dimensions of the doll house.

Since the scale factor is 1:10, this means that for every 1 inch on the drawing, the actual measurement is 10 inches. So, to find the actual length and width of the doll house, we can multiply the measurements on the drawing by 10.

7 1/5 inches * 10 = 71 inches + 2 inches

4 4/5 inches * 10 = 44 inches + 8 inches

So, the actual length of the doll house is 71 inches + 2 inches = 73 inches, and the actual width is 44 inches + 8 inches = 52 inches.

To find the area of the doll house, we can multiply the length and width:

73 inches * 52 inches = 3788 square inches

To convert this to square feet, we can divide by 12^2, since 1 foot = 12 inches:

3788 square inches / (12 inches)^2 = 31.5 square feet (rounded to the nearest square foot)

Hence, the area of the Temasa's doll house is 31.5 square feet.

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The described sampling method is an example of random sampling.

The sampling method described in the question is known as simple random sampling, which is a common method of selecting a representative sample from a population.

In this method, each member of the population is assigned a unique number, and a certain number of individuals are selected randomly from the population.

Simple random sampling is a type of probability sampling, which means that each member of the population has an equal chance of being selected for the sample.

This is important because it ensures that the sample is representative of the population as a whole, and reduces the risk of sampling bias.

The process of simple random sampling is straightforward and easy to implement, making it a popular choice for researchers. However, it can be time-consuming and may not be practical for very large populations. In such cases, other sampling methods such as stratified sampling or cluster sampling may be more appropriate.

Overall, simple random sampling is a reliable and effective method of selecting a representative sample from a population, and is widely used in research studies and surveys.

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Therefore, The Coordinates of the THIRD VERTEX is:  ( 5, -3 )

        Calculate the midpoint of the given vertices:

        MidPoint = ( -5 + 5/2,  3 + 3/2 )

        MidPoint = ( 0, 3 )

        Distance = √( -5 -5 )^2 + ( 3 - 3 )^2

        Distance = √( -10 )^2  +  (0)^2

        Distance = √100

        Distance =  10

        Side Length =  10/√3

        Side Length = 10/1.7

        Side Length = 5.88

        Height = √3/2  *  Side Length

        Height  = 1.7/2  *  5.88

        Height  =  5

Since the origin lies inside the triangle, The Third Vertex will have a Positive  X-Coordinate and a Negative Y-Coordinate.

       ( x, y )

        x  =  -5 + x/2

        y   =  3 + y/2

       x  =  5

       y  = -3

        Hence, The Coordinate  of the Third Vertex is:  ( 5, -3 )

I hope this helps!

The forecast for month 11 using simple exponential smoothing with α=.40 is 94.

To compute the forecast for month 11 using the exponentially smoothed forecast with α=.40, we need a time series dataset that includes the values of the variable we want to forecast for the past months. Exponential smoothing is a widely used time series forecasting method that works by giving more weight to recent observations while decreasing the weight of older observations in a weighted average.

In simple exponential smoothing, the forecast for the next period is computed as a weighted average of the actual value for the current period and the forecast for the previous period .

The weights decrease exponentially as we move back in time.

The smoothing parameter α controls the rate at which the weights decrease and the level of smoothing applied to the data.

A higher value of α puts more weight on recent observations and results in a more responsive forecast.

Assuming we have a time series dataset with values for months 1 through 10, we can use the following formula to compute the forecast for month 11 using simple exponential smoothing with α=.40:

F(t+1) = α×Y(t) + (1 - α) × F(t)

F(t+1) is the forecast for the next period (month 11), Y(t) is the actual value for the current period (month 10), and F(t) is the forecast for the current period (month 10).

Assuming the actual value for month 10 is 100 and the forecast for month 10 using the same method was 90, we can calculate the forecast for month 11 as:

F(11) = 0.4 × 100 + 0.6 × 90 = 94

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The distance between the two ships is 3.07 miles (approx). The given polar coordinates are converted into rectangular coordinates with the help of sine and cosine functions.

Given data:

The location of two ships from mays landing lighthouse, given in polar coordinates, are 3 mi, 170 and 5 mi, 150.

.To find:Distance between the ships

Formula used:

Distance between the ships = \(sqrt(d1^2 + d2^2 - 2*d1*d2*cos(theta1 - theta2)).\)

where d1 = 3 mi, theta1 = 170°, d2 = 5 mi, theta2 = 150°.

Calculation:Squaring and adding the given distances,sqrt(3² + 5² - 2*3*5*cos(170° - 150°))

:Distance between the ships is 3.07 miles (approx).

:Thus, the distance between the two ships is 3.07 miles (approx). The given polar coordinates are converted into rectangular coordinates with the help of sine and cosine functions. The formula used for finding the distance between the two ships is \(sqrt(d1^2 + d2^2 - 2*d1*d2*cos(theta1 - theta2)).\)

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Answer:

C

Step-by-step explanation:

hope this helps.

The statement that correctly compares the function shown on this graph with the function y=4x + 2 is 'The function shown on the graph has a smaller rate of change, but a higher starting point.'

"It is the rate at which one quantity is changing with respect to another quantity."

"y = mx + c, where m is the slope and c is the y-intercept"

For given question,

We have been given a function y = 4x + 2

This function represents a line.

Comparing above function with slop-intercept form of the line.

We have slope of the line = 4 and y - intercept = 2

We know that the rate of change of line is given by its slope.

So, the rate of change of given function y = 4x + 2 is 4.

A starting point of a function is the beginning value in a linear function.

The starting point of above function f(x) is 2.

The graph of the line passing through points (-1, 1) and (0, 4)

So using slope formula,

\(m1=\frac{4-1}{0-(-1)}\\\\ m1=\frac{3}{1}\\\\ m1=3\)

So, the rate of change of function shown on the graph is 3.

This means, the rate of change of function shown on the graph is smaller than the rate of change of function y = 4x + 2

The equation of the function shown on the graph would be y = 3x + 4

The starting point of function shown on the graph is 4.

This means the starting point of function shown on the graph is higher than the starting point of a function y = 4x + 2

Therefore, the statement that correctly compares the function shown on this graph with the function y=4x + 2 is 'The function shown on the graph has a smaller rate of change, but a higher starting point.'

The correct answer is option (A)

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Answer:

The answer would be 5.

Step-by-step explanation:

If you take away 18 from 90 would be 72.

72 - 18 =54

54 - 18 =36

36 - 18 =18

18 - 18 =0

If you count that would equal 5 .

Hope this helps .

The p-value can be calculated to test the null hypothesis that the average running time of HP laptops is 120 minutes against the alternative hypothesis that it is not equal to 120 minutes.

The p-value for testing the hypothesis that the average time for the HP laptops is not equal to 120 minutes can be calculated using a t-test. Given a sample mean of 122 minutes, a sample size of 60, a null hypothesis mean of 120 minutes, and a standard deviation of 25 minutes, we can calculate the t-value and find the corresponding p-value.

To calculate the t-value, we use the formula: t = (sample mean - null hypothesis mean) / (sample standard deviation / sqrt(sample size))

Plugging in the values, we get: t = (122 - 120) / (25 / sqrt(60))

Calculating the t-value, we find t ≈ 0.894

To find the p-value associated with this t-value, we can refer to a t-distribution table or use statistical software. The p-value represents the probability of observing a t-value as extreme as the one obtained, assuming the null hypothesis is true.

Since the p-value (0.757) is greater than the commonly used significance level of 0.05, we fail to reject the null hypothesis. This suggests that there is not enough evidence to conclude that the average running time of HP laptops is significantly different from 120 minutes.

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Therefore, you traveled a distance of 10080 meters (or 10.08 kilometers) while jogging at a speed of 2.8 m/s for one hour.

To find the distance you traveled while jogging at a speed of 2.8 m/s for an hour, you can use the formula:

Distance = Speed * Time

Given:

Speed = 2.8 m/s

Time = 1 hour

Plugging in these values into the formula, we have:

Distance = 2.8 m/s * 1 hour

However, we need to convert the time from hours to seconds, as the speed is given in meters per second. There are 3600 seconds in an hour.

Distance = 2.8 m/s * (1 hour * 3600 seconds/hour)

Simplifying the expression, we get:

Distance = 2.8 m/s * 3600 seconds

Calculating this expression, we find:

Distance = 10080 meters

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The conditions for inference are not met. Specifically, the random condition is not met for both samples.

For inference to be valid, the samples should be selected randomly from their respective populations. In this case, the coach selected a random sample of 5 players from his team and 5 players from the rival team. However, since the selection was limited to only 5 out of 11 players in each team, the random condition is not fully satisfied.

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Answer:

The probability that the socks picked is not blue is 11/14

Step-by-step explanation:

In this question, we are to calculate that a socks selected from random from the group of socks given in the question is not a blue socks.

This means the socks would either be a black or a white socks

Firstly, we find the total number of socks = 8 + 6 + 14 = 28 socks

Now the probability of picking a blue socks will be 6/28 = 3/14

We must understand that the probability of not picking a blue socks = 1 - the probability of picking a blue socks

Thus, the probability of not picking a blue socks will be 1 - 3/14 = 11/14

Answer:

The probability that a sock chosen at random is not blue P = 11/14 = 0.786

Step-by-step explanation:

Probability is the chances of getting a particular outcome.

Given:

Black socks = 8

Blue socks = 6

White socks = 14

Total = 8+6+14 =28

The probability that a sock chosen at random is not blue P;

P = number of non-blue socks ÷ total number of socks

Number of non blue socks = black + white socks = 8+14 = 22

Substituting the values;

P = 22/28 = 11/14

P = 11/14 = 0.786

Answer:

 2.5L of 5% saline and 7.5L of 20% saline solution

Step-by-step explanation:

The length of BP is 14.4 meters.

To find the length of BP, we can use the property that states that when two chords intersect inside a circle, the product of the segment lengths on one chord is equal to the product of the segment lengths on the other chord.

Using this property, we can set up the equation:

AP * PC = BP * DP

Substituting the given values:

12 m * 6 m = BP * 5 m

Simplifying:

72 m^2 = BP * 5 m

To solve for BP, divide both sides of the equation by 5 m:

72 m^2 / 5 m = BP

Simplifying:

14.4 m = BP

Therefore, the length of BP is 14.4 meters.

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Answer:A

Step-by-step explanation:  divide both by 5 and you get 3/4 because 15 divided by 5 = 3 and 20 divided by 5 =4

Answer:  see proof below

Step-by-step explanation:

Use the following Sum to Product Identities:

sin (A) - sin (B) = 2 cos (A+B)/2 · sin (A-B)/2

sin (A) + sin (B) = 2 sin (A+B)/2 · cos (A-B)/2

Given:  sin Ф = k sin β      -->   (sin Ф)/(sin β) = k

Proof RHS → LHS

\(\text{RHS:}\qquad \qquad \qquad \dfrac{k-1}{k+1}\tan\dfrac{\theta+\beta}{2}\)

\(\text{Given:}\qquad \qquad \dfrac{\frac{\sin \theta}{\sin \beta}-1}{\frac{\sin \theta}{\sin \beta}+1}}\cdot \tan\dfrac{\theta +\beta}{2}\)

\(\text{Simplify:}\qquad \qquad \dfrac{\frac{\sin \theta}{\sin \beta}-\frac{\sin \beta}{\sin \beta}}{\frac{\sin \theta}{\sin \beta}+\frac{\sin \beta}{\sin \beta}}}\cdot \tan\dfrac{\theta +\beta}{2}\\\\\\.\qquad \qquad \qquad = \dfrac{\sin \theta -\sin \beta}{\sin \theta +\sin \beta}\cdot \tan\dfrac{\theta +\beta}{2}\)

\(\text{Product to Sum:}\qquad \quad \dfrac{2\cos \frac{\theta+\beta}{2}\cdot \sin \frac{\theta-\beta}{2}}{2\sin \frac{\theta+\beta}{2}\cdot \cos \frac{\theta-\beta}{2}}\cdot \tan\dfrac{\theta +\beta}{2}\)

\(\text{Expand:}\qquad \qquad \dfrac{2\cos \frac{\theta+\beta}{2}\cdot \sin \frac{\theta-\beta}{2}}{2\sin \frac{\theta+\beta}{2}\cdot \cos \frac{\theta-\beta}{2}}\cdot \dfrac{\sin\frac{\theta +\beta}{2}}{\cos \frac{\theta +\beta}{2}}\)

\(\text{Simplify:}\qquad \qquad \quad \quad \dfrac{\sin \frac{\theta +\beta}{2}}{\cos \frac{\theta+ \beta}{2}}\\\\\\.\qquad \qquad \qquad \quad =\tan\dfrac{\theta +\beta}{2}\)

 \(\text{LHS = RHS:}\quad \tan\dfrac{\theta +\beta}{2}=\tan\dfrac{\theta +\beta}{2}\quad \checkmark\)

Answer:

  y = -3√(x -1) +2

Step-by-step explanation:

You want the equation of the translated, reflected, and scaled square root function in the given graph.

We can find the amount of translation by looking at the position of the point that corresponds to (0, 0) on the graph of the parent function. That end point where the slope is vertical is located at (1, 2) on the given graph.

When a function y = f(x) is translated by (h, k), it becomes ...

  y = f(x -h) +k

For (h, k) = (1, 2) the translated function is ...

  y = f(x -1) +2 = √(x -1) +2

When the function is scaled, its value is multiplied by the scale factor. If that factor is 'a', our scaled and translated function is ...

  y = a√(x -1) +2

To find the scale factor, we can use the second point on the curve: (2, -1). Using these values for x and y gives ...

  -1 = a√(2 -1) +2

  -3 = a . . . . . . . . . . subtract 2, simplify

The transformed function is ...

  y = -3√(x -1) +2

__

Additional comment

The function is scaled before it is translated, so the scale factor does not apply to the translation.

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Although the rating score is now less important compared to dollar income, strategy A still yields the highest payoff in terms of D+R for both citizens.

The equilibrium outcome remains unchanged, and both citizens will still choose strategy A.

(b) In this new game where citizens care about both their dollar earnings and the rating score, we can construct a payoff matrix by adding the dollar income and the rating score for each citizen.

Let's denote the dollar income as "D" and the rating score as "R".

Assuming the original payoff matrix represents the dollar income, we can add the rating scores to each entry:

A

B

C

A

8+8=16

6+6=12

0+0=0

B

6+6=12

4+4=8

0+0=0

C

0+0=0

0+0=0

0+0=0

In this new game, the equilibrium outcome (choice of strategies) would still be for both citizens to choose strategy A.

By choosing A, each citizen maximizes their dollar income (D) as well as the rating score (R) since A yields the highest payoff in terms of D+R for both citizens.

Therefore, the equilibrium outcome is for both citizens to choose strategy A.

(c) If each player treats the rating score as equivalent to 50 cents earned, we need to adjust the payoff matrix accordingly by multiplying the rating scores by 0.5:

A

B

C

A

8+4=12

6+3=9

0+0=0

B

6+3=9

4+2=6

0+0=0

C

0+0=0

0+0=0

0+0=0

In this case, the equilibrium outcome would still be for both citizens to choose strategy A.

Although the rating score is now less important compared to dollar income, strategy A still yields the highest payoff in terms of D+R for both citizens.

Therefore, the equilibrium outcome remains unchanged, and both citizens will still choose strategy A.

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