4. Corrupt professor Z has a class of 50 students. He needs to give exactly 10 A's. However five students already have a special deal (they are professor Z's nephews and nieces) and will get A's for sure. How many ways can the 10 A's be distributed?

Answer:

x = 10

Step-by-step explanation:

-36 = -6(2x-14)

6 = 2x-14

20 =2x

10 = x

Answer:

-x = -10

Step-by-step explanation:

-36 = -12x + 84

-120 = -12x

-10 = -x

Answer:

The answer to your question is: d = 12

Step-by-step explanation:

                                             6(d+1)−2d=54

     Expand                          6d + 6 -2d = 54

                                            6d - 2d = 54 - 6

     Simplify                          4d = 48

                                             d = 48 / 4

        Result                           d = 12

999 is 3 times 333, so the ratio will have to be multiplied by three.

No of cherries needed = 240240240 * 3 = 720720720 cherries

Answer:

An equation that expresses a relationship between two or more​ variables, such as Upper H equals nine tenths left parenthesis 220 minus a right parenthesis ​, is called​ Formula.

A formula is an equation that expresses a relationship between two or more variables.

The process of finding formulas to describe real-world phenomena is called mathematical modeling.

Such​ equations, together with the meaning assigned to the​ variables, are called mathematical​ models.

A mathematical equation expresses a relationship between variables, and the process of finding such equations is called mathematical modeling.

An equation that expresses a relationship between two or more variables is called a mathematical equation. The process of finding such equations to describe real-world phenomena is called mathematical modeling. These equations, together with the meaning assigned to the variables, are called mathematical models.

For example, if we have an equation like Upper H equals nine tenths left parenthesis 220 minus a right parenthesis, this equation represents a relationship between a variable H and the expression nine tenths left parenthesis 220 minus a right parenthesis. We can use this equation to calculate the value of H given a specific value for the expression.

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Answer:

24/18=1.333

20/16=1.25

6/4=1.5 (the ratio to achieve9

1.333 is more close to 1.5 than 1.25 (11% difference compared to 17%)

Step-by-step explanation:

Answer:

16 in. X 20 in will keep more.

Step-by-step explanation:

Length of picture = 4 inches

Breadth of picture = 6 inches

Area of picture=[tex]length  \times breadth[/tex]

                        =[tex]4 \times 6[/tex]

                        =[tex]24 inches^2[/tex]

Length of frame 1 = 16 inches

Breadth of frame 1 = 20 inches

Area of frame 1 = [tex]16 \times 20 = 320 inches^2[/tex]

So, % of picture can fit in frame 1= [tex]\frac{\text{original picture area }}{\text{Frame 1 area }} \times 100[/tex]

                                                   = [tex]\frac{24}{320} \times 100[/tex]

                                                   = [tex]7.5 %[/tex]

Length of frame 2 = 18 inches

Breadth of frame 2 = 24 inches

Area of frame 2 = [tex]18 \times 24 = 432 inches^2[/tex]

So, % of picture can fit in frame 2 = [tex]\frac{\text{original picture area }}{\text{Frame 1 area }} \times 100[/tex]

                                                   = [tex]\frac{24}{432} \times 100[/tex]

                                                   = [tex]5.56 %[/tex]

Since % of picture can fit in frame 1 is more than frame 2 .

So, 16 in. X 20 in will keep more.

Answer: no its too big it wont fit

Step-by-step explanation:

its just too big

Answer:

  2.592×10^5 seconds

Step-by-step explanation:

[tex]3\,days\cdot\dfrac{24\,h}{1\,day}\cdot\dfrac{3.6\cdot 10^3\,s}{1\,h}=3\cdot 24\cdot 3.6\cdot 10^3\,s=2.592\cdot 10^5\,s[/tex]

Answer:

Corrected answer is 8.64 x 10^4 seconds in one day

Step-by-step explanation:

Answer:

Using backward reasoning we want to show that "We can never get nine 0's".

Step-by-step explanation:

Basically in order to create nine 0's, the previous step had to have all 0's or all 1's. There is no other way possible, because between any two equal bits you insert a 0.

If we consider two cases for the second-to-last step:

There were 9 0's:

We obtain nine 0's if all bits in the previous step were the same, thus all bit were 0's or all bits were 1's. If the previous step contained all 0's, then we have the same case as the current iteration step. Since initially the circle did not contain only 0's, the circle had to contain something else than only 0's at some point and thus there exists a point where the circle contained only 1's.

There were 9 1's:

A circle contains only 1's, if every pair of the consecutive nine digits is different. However this is impossible, because there are five 1's and four 0's (we have an odd number of bits!), thus if the 1's and 0's alternate, then we obtain that 1's that will be next to each other (which would result in a 1 in the next step). Thus, we obtained a contradiction and thus assumption that the circle contains nine 0's after iteratins the procedure is false. This then means that you can never get nine 0's.

To summarize, in order to create nine 0's, the previous step had to have all 0's or al 1's. As we didn't start the arrange with all 0's, the only way is having all 1's, but having all 1's will not be possible in our case since we have an odd number of bits.

(D). 8 Wine, 16 Cloth

Cloth production by Argentina = 20 units

Wine production by Argentina = 2 units

Cloth production by Chile = 24 units

Wine production by Chile = 12 units

Now,

2 units of labor was transferred by Argentina from Wine to Cloth and 1 unit of labor was transferred by Chile from Cloth to Wine.

Therefore,

Increase in Cloth production = Argentina's total cloth produced - Cloth produced by Chile (before transfer)

Now,

We can say that total cloth produced by Argentina is = 2 x 20 = 40 units

So,

Increase in Cloth production = 40 - 24 = 16 units

Therefore, the increase in Cloth Production is 16 units.

Similarly,

Increase in Wine production = Chile's total Wine produced - Wine produced by Argentina (before transfer)

Now,

We can say that total Wine produced by Chile is = 1 x 12 = 12 units

Wine produced by Argentina before transfer = 2 x 2 = 4 units

So,

Increase in Wine production = 12 - 4 = 8 units

Therefore, the increase in Wine Production is 8 units.

Hence, the correct option is (D).

Answer and Step-by-step explanation:

As quantifiers, we can settle:

x is a student

M(x) is a math major student

C(x) is a computer science major student

F(x) is a freshman student

S(x) is a sophomore student

J(x) is a junior student

N(x) is a senior student

∃ exists

∀ every

¬ negation

∧ and

∨ or

a) There is a student in the class who is a junior.

∃xJ(x) value: True. There are 4 juniors

b) Every student in the class is a computer science major.

∀xC(x) value: False. There are math students

c) There is a student in the class who is neither a mathematics major nor a junior.

∃x¬M(x)∨¬C(x) value: False. All students are math ou computer science majors

d) Every student in the class is either a sophomore or a computer science major.

∀xS(x)∨C(x) value: False. There are some students who are neither, for example mathematics majors who are juniors

e) There is a major such that there is a student*

∃M(x)C(x)x value: True. All majors have students.

*This one seems incomplete, but I answered the way it is writen.

The expression of the statement based on the quantifiers show that the truth value will be:

It should be noted that quantifies are the words or expressions that indicate the number of elements which a statement pertains to.

From the information, there is a student in the class who is a junior. It can also be deduced that not every student in the class is a computer science major. This is because there are mathematics majors too.

Furthermore, there is a student in the class who is neither a mathematics major not a junior but not every student in the class is either a sophomore or a computer science major.

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Answer: Ok, the chanche of both children were born on a monday is [tex]\frac{1}{7}[/tex]

Step-by-step explanation: Well, we alredy know that one of her children was born on a monday, when they ask the probability of both children were born on a monday, we only need to see the case of the second kid.

So, the week has 7 days, ence the probability for each day (in this case a monday) is 1/7.

15=3n+6p

We need to isolate n.

15 - 6p = 3n

(15 - 6p)/3 = n

5 - 2p = n

To solve for n in the equation 15 = 3n + 6p, isolate n by subtracting 6p from both sides, and then divide both sides by 3.

To solve for n in the equation 15 = 3n + 6p, we can start by isolating the variable n. First, subtract 6p from both sides of the equation to get 15 - 6p = 3n. Then, divide both sides by 3 to solve for n, yielding: n = (15 - 6p) / 3.

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Answer:

The answer to your question is:  BC = √65 or 8.05 u

Step-by-step explanation:

Data

AD = 12 u

BD = 15 u

AC = 4 u

BC = ?

First calculate the length of AB using the pythagorean theorem

 BD² = AB² + AD²

 AB² = BD² - AD²

 AB² = 15² - 12²

 AB² = 225 - 144

 AB² = 81

 AB = 9 u

Now, use the pythagorean theorem to find BC

 AB² = BC² + AC²

 BC² = AB² - AC²

 BC² = 9² - 4²

 BC²  = 81 - 16

 BC² = 65

 BC = √65 or 8.05 u

Answer:

50 minutes

Step-by-step explanation:

Given,

Water can be drained out of the tub at a rate of 2 gallons per minute,

So, the drained water in 1 minute = 2 gallon,

That is, change in 1 minute = -2 gallon

( negative sign shows losing water)

Also, it is filled by a faucet at a rate of 1 gallon per minute,

So, the filled water in 1 minute = 1 gallon,

That is, change in 1 minute = + 1 gallon

( Positive sign shows additional water ),

Thus, total change in 1 minute = -2 + 1 = -1 gallon,

Let x be the time after which the bathtub will be emptied completely,

Total change in x minutes = -x gallon,

Bathtub will be emptied, if,

Initial volume of water + total change in water = 0

50 - x = 0  ( given volume of tub is 50 gallon )

[tex]\implies x = 50[/tex]

Hence, it will take 50 minutes to drain the full tub.

Final answer:

To drain a bathtub initially filled with 50 gallons, with an incoming rate of 1 gallon per minute and a draining rate of 2 gallons per minute, it takes 50 minutes.

Explanation:

The question involves calculating the time it takes to drain a bathtub that is being filled and drained simultaneously. Initially, the bathtub is completely filled with 50 gallons of water. The faucet fills the tub at a rate of 1 gallon per minute, while the drain can remove water at a rate of 2 gallons per minute. Therefore, the net rate at which water is being drained is 1 gallon per minute (2 gallons out - 1 gallon in). To completely drain the tub of its initial 50 gallons, at a net rate of 1 gallon per minute, it would take 50 minutes.

Therefore, as per the above explaination,  the correct answer is 50 min.

Answer:

75%

Step-by-step explanation:

100%-25%= 75%

He would have paid 75% of the original price..

Answer:

C. [tex]\frac{3}{5}[/tex]

Step-by-step explanation:

Let x be the original price of laptop computer.

We have been given that Luis purchased a laptop computer that was marked down by 2/5 of the original price.

The price of laptop computer after mark-down would be x minus 2/5 of x.

[tex]\text{The price of laptop computer after mark-down}=x-\frac{2}{5}x[/tex]

[tex]\text{The price of laptop computer after mark-down}=\frac{5}{5}x-\frac{2}{5}x[/tex]

[tex]\text{The price of laptop computer after mark-down}=\frac{5-2}{5}x[/tex]

[tex]\text{The price of laptop computer after mark-down}=\frac{3}{5}x[/tex]

Therefore, Luis paid [tex]\frac{3}{5}[/tex] of the original price.

Answer:

  A. (1, -1)

  B. (1, -1), (2, 0)

  C. x = 0

Step-by-step explanation:

A graph is a plot of all the points that are solutions to an equation.

Part A:

A point will be a solution to two equations if it is a point of intersection of their graphs. The one point that is a solution to both p(x) and g(x) is the one point where their graphs intersect: the red and blue lines cross at (1, -1).

__

Part B:

Any other point on the graph p(x) will be another solution of it. One that is near to the point of intersection with g(x) is the point where p(x) crosses the x-axis: (2, 0). Of course, the solution listed in Part A is also a solution to p(x).

__

Part C:

The point where the graph of g(x) crosses the graph of f(x) is (0, 3). The x-value that makes g(x) = f(x) is x=0. That is the solution to this equation. (We don't really care what the values of f(0) and g(0) are--just that they are equal.)

Answer: Simple random sampling.

Step-by-step explanation:

A simple random sample is basically a subset (with size n) from the entire population, where the chance of getting selected for each element is equal.

Given :A research company believes teens today are getting less than the recommended hours of sleep. The company takes a list of 2000 teen volunteers and assigns each volunteer a number. A

A random number generator is used to select 350 individuals to take part in a sleep survey.

It is a simple random sampling because the researcher selected participants randomly such that the chance to get selected for each of them remains same.

Answer:

Option d) Chebyshev's rule

Step-by-step explanation:

The Chebyshev's rule state that for a data that is not distributed normally,

atleast [tex](1 - \frac{1}{k^2})\% \text{ of data lies within the interval}~(Mean \pm (k)Standard ~Deviation)[/tex].

Here, k cannot be 1 and is always greater than 2.

For k = 2,

[tex](1 - \frac{1}{4})\times 100\% = 75\%[/tex] of data lies within the range of [tex](\mu \pm 2\sigma)[/tex]

Atleast 75% of children finished their vegetables in [tex](\mu \pm 2\sigma) = (4.2 \pm (2)1.0) = (2.2,6.2)[/tex]

For k = 3,

[tex](1 - \frac{1}{9})\times 100\% = 88.912\%[/tex] of data lies within the range of [tex](\mu \pm 3\sigma)[/tex]

Atleast 89% of children finished their vegetables in [tex](\mu \pm 3\sigma) = (4.2 \pm (3)1.0) = (1.2,7.2)[/tex]

Thus, option d) is correct.

Answer:

The answer to your question is:

a) C = 2g + 155c

b) g = 25 grapes

c) c = 3

Step-by-step explanation:

Data

grapes = g = 2 calories

cheese = c = 155 calories

a) Equation, we consider the amount of grapes and the calores given.

Total calories = C = 2g + 155c

b) We consider that the slices of cheese stays the same

                       2g + 155 = 205

                       2g = 205 -155

                        2g = 50

                        g = 50/2 = 25 grapes

c) Then the number of grapes stays the same

                       2(25) + 155c = 515

                       50 + 155c = 515

                       155c = 515 - 50

                        155c = 465

                        c = 465/155

                        c = 3 slices of cheese

Answer: 180

Step-by-step explanation:

divide 45 by 7.5 to get the amount of dollars earned per hour

45/7.5 = 6

6(x)= 30hours, = 6(30) = $180

Answer:

$180

Step-by-step explanation:

let pay be p and hours worked be h

Given p varies directly as h then the equation relating them is

p = kh ← k is the constant of variation

To find k use the condition p = 45 when h = 7.5, then

k = [tex]\frac{p}{h}[/tex] = [tex]\frac{45}{7.5}[/tex] = 6, thus

p = 6h ← equation of variation

When h = 30, then

p = 6 × 30 = $180

Answer:

Option c: all real numbers greater than 0

Step-by-step explanation:

we have

[tex]f(x)=\frac{1}{7}(9^{x})[/tex]

This is a exponential function of the form

[tex]f(x)=a(b^{x})[/tex]

where

a is the initial value (y-intercept)

b is the base

r is the rate

b=(1+r)

In this problem we have

a=1/7

b=9

r=b-1 ----> r=9-1=8 -----> r=800%

using a graphing tool

see the attached figure

The domain is the interval ------> (-∞,∞)

The domain is all real numbers

The range is the interval ---------> (0,∞)

The range is all real numbers greater than zero

Answer: all real numbers greater than 0

Step-by-step explanation:

Range is the set of y values for which the function is defined                    using a graphing tool

The domain is the interval ----> (-∞,∞) All real numbers

For all positive and negative values for x the value of y is always positive

The range is the interval ---->(0,∞)

All real numbers greater than 0

Answer:

Option 3) We reject the null hypothesis with one tail test and accept the null hypothesis with two tail test.

Step-by-step explanation:

We are given the following information:

n = 16

[tex]t_{statistic} = 1.94[/tex]

[tex]\alpha = 0.05[/tex]

Now,

Right One-tail Test

[tex]t_{critical} \text{ at 0.05 level of significance, 15 degree of freedom } = 1.753[/tex]

[tex]t_{stat} > t_{critical}[/tex]

We reject the null hypothesis in this case.

Two-tail Test

Now, [tex]t_{critical} \text{ at 0.05 level of significance, 9 degree of freedom } = \pm 2.131[/tex]

[tex]-2.131 < t_{stat} < 2.131[/tex]

We accept the null hypothesis in this case.

Option 3) We reject the null hypothesis with one tail test and accept the null hypothesis with two tail test.

The correct decision for this hypothesis test is to reject the null hypothesis with a one-tailed test but fail to reject with two tails.

To make a decision in a hypothesis test, we compare the t statistic to the critical value. In this case, the t statistic is 1.94. Since the treatment is expected to increase scores and the sample mean shows an increase, we are conducting a one-tailed test. Looking at the critical value for a = 0.05 for a one-tailed test using the t15 distribution, we find that it is 1.753. Since the t statistic (1.94) is greater than the critical value (1.753), we reject the null hypothesis. Therefore, the correct decision for this hypothesis test is to reject the null hypothesis with a one-tailed test but fail to reject with two tails.

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Answer:

the total charge is

[tex]Q=24(1-\exp(-1))nC\approx15.171nC[/tex]

Step-by-step explanation:

Since x is measured from the center, that means that x=0 is the center so the edges of the rod correspond to x=-6 and x=6. that meas that the total charge can be calculated as

[tex]Q=\int^{6}_{-6}2\exp\left(\frac{-|x|}{6}\right)dx[/tex]

separating the integral from -6 to 0 and from 0 to 6, taking into account that |x|=-x for x<0 and |x|=x for x >=0, we get[tex]Q=\int^{0}_{-6}2\exp\left(\frac{x}{6}\right)dx+\int^{6}_{0}2\exp\left(\frac{-x}{6}\right)dx[/tex]

using the substitution x=-u in the first integral we get[tex]\int^{0}_{-6}2\exp\left(\frac{x}{6}\right)dx=-\int^{0}_{6}2\exp\left(\frac{-u}{6}\right)du=\int^{6}_{0}2\exp\left(\frac{-u}{6}\right)du[/tex]

which is the same as the first integral. Thus, the total charge is given by

[tex]Q=2\int^{6}_{0}2\exp\left(\frac{-x}{6}\right)dx[/tex]

integrating we get

[tex]Q=4(-6\exp\left(\frac{-x}{6}\right))\big|^{6}_{0}=-24(\exp(-6/6)-\exp(0))=24(1-\exp(-1))[/tex]

The total charge is Q= 15.171nC.

Since x is measured from the center, that means that x=0 is the center.

So, the edges of the rod correspond to

x=-6 and x=6.

That means that the total charge can be calculated as

[tex]Q= \int\limits^6_ 6 2 exp(-|x|/6)dx[/tex]

separating the integral from -6 to 0 and from 0 to 6,

Taking into account that

|x|=-x for x<0

and |x|=x for x >=0

Thus, the total charge is given by:

[tex]Q= 2\int\limits^6_0 2exp (-x/6), dx[/tex]

When we integrate, we get:

Q= 24(1- exp(-1))nC ≈

15.171nC

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Answer:

The first customer that will get four free tickets is 50th customer

Step-by-step explanation:

Find the least common multiple of numbers 10 and 25. First, factorize these numbers:

[tex]10=2\cdot \underline{5}\\ \\25=\underline{5}\cdot 5\\ \\LCM(10,25)=\underline{5}\cdot 2\cdot 5=50[/tex]

When finding LCM, first write the all common multiples (underlined 5) and then multiply them by remaining multiples (2 and 5). You get 50 as LCM(10,25). This means that each 50th customer will get four free tickets.

Final answer:

The first customer who will receive four free tickets is the 100th customer.

Explanation:

To determine the first customer who will receive four free tickets, we need to find the smallest positive integer that is divisible by both 10 and 25. This is called the least common multiple (LCM). To find the LCM of 10 and 25, we can list the multiples of each number until we find a common multiple:

Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100

Multiples of 25: 25, 50, 75, 100

The LCM of 10 and 25 is 100. Therefore, the first customer who will receive four free tickets is the 100th customer.

Answer:

width of a singles court = 27 feet

Step-by-step explanation:

Width of a doubles tennis court = 36 feet

Width of a singles tennis court = 75% of 36

[tex]width = 75\% \times 36 \\ width = \frac{75}{100} \times 36 \\ width = \frac{3} {4} \times 36 \\ width = \frac{108}{4} = 27[/tex]

The percentage of men who gave an excellent rating is 70%. The percentage of women who gave an excellent rating is 64%. The total percentage of customers giving an excellent rating is 67%.

To find the percentage of men who gave an excellent rating, we divide the number of men who gave an excellent rating (175) by the total number of men surveyed (250) and multiply by 100.

So the percentage of men who gave an excellent rating is 70%.

To find the percentage of women who gave an excellent rating, we divide the number of women who gave an excellent rating (160) by the total number of women surveyed (250) and multiply by 100.

So the percentage of women who gave an excellent rating is 64%.

To find the total percentage of customers who gave an excellent rating, we divide the total number of customers who gave an excellent rating (175 + 160 = 335) by the total number of customers surveyed (500) and multiply by 100. So the total percentage of customers who gave an excellent rating is 67%.

The percentage of men who gave an excellent rating is 70%, the percentage of women who gave an excellent rating is 80%, and the total percentage of customers giving an excellent rating is 67%.

First, let's calculate the percentage of men who gave an excellent rating:

- There are 250 men surveyed.

- Out of these, 175 men rated the customer service as excellent.

- To find the percentage, we use the formula: (Number of men who rated excellent / Total number of men) × 100.

- Plugging in the numbers, we get: (175 / 250) × 100.

- Simplifying this, we divide both the numerator and the denominator by 25 to get: (7 / 10) × 100.

- This simplifies to 70%.

Next, we calculate the percentage of women who gave an excellent rating:

- There are 250 women surveyed.

- Out of these, 160 women rated the customer service as excellent.

- Using the same formula as before: (Number of women who rated excellent / Total number of women) × 100.

- Plugging in the numbers, we get: (160 / 250) × 100.

- Simplifying this, we divide both the numerator and the denominator by 40 to get: (4 / 5) × 100.

- This simplifies to 80%.

Finally, we calculate the total percentage of customers who gave an excellent rating:

- The total number of customers surveyed is 500 (250 men + 250 women).

- The total number of excellent ratings is 175 from men and 160 from women, which sums up to 335.

- Using the formula: (Total number of excellent ratings / Total number of customers) × 100.

- Plugging in the numbers, we get: (335 / 500) × 100.

- Simplifying this, we divide both the numerator and the denominator by 5 to get: (67 / 100) × 100.

- This simplifies to 67%.

- Percentage of men giving an excellent rating: 70%

- Percentage of women giving an excellent rating: 80%

- Total percentage of customers giving an excellent rating: 67%

Answer:

The answer to your question is: 99 m²

Step-by-step explanation:

Data

height = 11 m

area of the square base = 9 m²

Formula

Volume of a right rectangular prism = area of the base x height

                                                           = 11 x 9     substitution

                                                           = 99 m²

Answer:

Step-by-step explanation:

From the problem statement, each box of candles has the following range of candles:

[tex]78 \leq x \leq 82[/tex]

We also know that we have 50 boxes of candles, so we multiply the above range by 50 to get the range of candles:

[tex]3900 \leq x \leq 4100[/tex]

Finally, each candle costs $2, so we have the final range of cost:

[tex]7800 \leq x \leq 8200[/tex]

By calculating the cost of producing the minimum and maximum number of candles that can be packaged in 50 boxes, we determine the range of possible production costs, x, is $7,800 to $8,200.

The student needs to calculate the range of possible production costs for 50 boxes of candles, given that each box contains an average of 80 candles and the number of candles can vary by at most two from this average. Since each candle costs two dollars to produce, we can find the minimum and maximum number of candles in one box by subtracting and adding two to the average, respectively (78 and 82 candles). Multiplying these numbers by the cost per candle gives us the cost per box, and then multiplying by the number of boxes (50) gives us the total production cost range for all boxes.

To calculate the minimum cost, we use the minimum number of candles per box: 78 candles per box × $2 per candle × 50 boxes = $7,800. To calculate the maximum cost, we use the maximum number of candles per box: 82 candles per box × $2 per candle × 50 boxes = $8,200. Therefore, the range for the possible production costs, x, for 50 boxes of candles is $7,800 ≤ x ≤ $8,200, which corresponds to answer choice D.

Answer:

  y = x + 4

Step-by-step explanation:

The line of reflection is the perpendicular bisector of segment AA', so passes through point (A+A')/2 = (-2, 2) and is perpendicular to the line through A and A'. That line is y = -x, so the point-slope equation of the line of reflection is ...

  y = 1(x -(-2)) +2

  y = x +4

The line of reflection between triangle ABC and its image A'B'C' is y = -x. The point-slope equation of the line of reflection is y = x+4.

To find the line of reflection between triangle ABC and its image A'B'C', we can observe that the corresponding points have the same x-coordinates and their y-coordinates are negatives of each other. Since the line of reflection is the perpendicular bisector of the segment joining each original point and its image, we can use the coordinates of two corresponding points to find the equation of the line. In this case, we can use points A and A', and points B and B' to determine the line of reflection.

Using the coordinates A(-4, 4) and A'(0, 0), we can calculate the slope of the line as (0 - 4) / (0 - (-4)) = -1. The midpoint between A and A' is (-2, 2), which lies on the line. So, the equation of the line is y = -x.

Similarly, using the coordinates B(-4, 1) and B'(-3, 0), we can calculate the slope as (0 - 1) / (-3 - (-4)) = 1. The midpoint between B and B' is (-3.5, 0.5), which also lies on the line y = x. Therefore, the line of reflection is y = -x.

So, the point-slope equation of the line of reflection is:

 y = 1(x -(-2)) +2

 y = x +4

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The graph for reflection is given below: