5. Write an expression for the area of a rectangle with length & and width w.

6. Imagine dilating the rectangle with length & and width w by a factor of k.

7. Write an expression for the area of the dilated rectangle.

8. Use your work to draw a conclusion about what happens to the area of a rectangle when it's dilated by a scale factor of k.

The solution is, it represents a vertical translation, and value is (d)  k = 8.

Transformations are changes done in the shapes on a coordinate plane by rotation, reflection or translation.

here, we have,

Translations of various kinds are often most easily seen by comparing points where the graph has a distinctive feature. On this graph, that is the vertex.

When looking at the graph we can discard 2 of the options, these are rotation (as they are parallel, it is impossible to find a rotation axis) and reflection. We can observe that it occur a translation. To know what kind of translation happened, let's look at the y intercept of both lines.

For f(x) the y intercept is 0 and for g(x) the y intercept is -4, it means it occured a vertical translation down 4 units.

so, we have,

The value k in the equation ...

 g(x) = f(x) +k

represents a vertical translation by k units.

The vertex coordinates for f(x) and g(x) are ...

 f(x): vertex = (-3, -3)

 g(x): vertex = (-3, 5)

This means ...

 f(-3) = -3

 g(-3) = 5 = f(-3) +k = -3 +8  

⇒   k = 8

Hence, The solution is, it represents a vertical translation, and value is (d)  k = 8.

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The parallel line pairs, AB and BC ; CE and AD are present in the above figure. The sum of the angles a+b+c+d+e without measuring the angles is equals to the 180°.

We have the lines, AB and BC with arrows in above figure are parallel and we determine the sum of the angles a + b + c + d + e, without measuring the angles. Both of lines, AB and BC intersects each other at point B. Similarly, AD and CE lines also intersect. Properties of parallel lines are

Measure of angle, ECD = d

Measure of angle, CED = c

Measure of angle, ADE = b

Measure of angle, DAE = e

Measure of angle CBA = a ( corresponding angles)

Now, measure of angle ACE, ∠ACE= ∠CED = c ( alternating angles)

Similarly, ∠CAE = ∠CDE = e (alternating angles)

Now, measure of angle ACB = measure of angle BCE + measure of angle ACE

= c + d

Measure of angle CAB = measure of angle CAD + measure of angle DAB

= e + b

Sum of interior angles of a triangle is equals to the 180°. So, ∠ACB + ∠ABC + ∠BAC = 180°

=> c + d + e + b + a = 180°

Hence, required sum of angles is 180°.

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Complete question:

Given that the lines with arrows in above figure are parallel , determine the sum of the angles a+b+c+d+e without measuring the angles. Explain your reasoning

The function that models the total monthly cost, based on the data in the table is; f(x) = 1.25·x + 30

A function is a rule, definition or law that specifies an output based on the input provided , such that each input is mapped to exactly one output

The function that models the monthly cost can be found as follows;

Number of songs downloaded [tex]{}[/tex]       Monthly cost

0 [tex]{}[/tex]                                                            30.00

1[tex]{}[/tex]                                                              31.25

2[tex]{}[/tex]                                                             32.50

3[tex]{}[/tex]                                                             33.75

4 [tex]{}[/tex]                                                            35.00

The required value of x is; x ≥ 0

The function, f(x) can be found by analyzing the data provided in the table as follows;

The first difference of the data in the Monthly cost column is; 35 - 33.75 = 33.75 - 32.5 = 32.5 - 31.25 = 31.25 - 30.00 = 1.25 = Constant

The (first) difference in the values in the Number of Songs Downloaded column is 1 (constant), therefore, the function is a linear function with a slope of 1.25

The value of the function which is the Monthly cost When the Number of Songs Downloaded is 0 is 30, therefore, the y-intercept is; c = 30

The function f(x) is therefore; f(x) = 1.25·x - 30

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The sets of points that have a distance of 4 units between them are:

Point X and Point W

Point T and Point S

To find the distance between two points, we use the distance formula:

distance = √ (x₂ – x₁)² + (y₂ – y₁) ²

We can calculate the distances between each pair of points and check if any of them are equal to 4.

The distances between the following pairs of points are 4 units:

Point X (3,9) and Point W (5,0): distance = √((5 - 3)² + (0 - 9)²) = √52 = 4.12

Point T (5,0) and Point S (-1,0): distance = √((-1 - 5)² + (0 - 0)²) = √36 = 6

Point T (5,0) and Point S (-3,0): distance = √((-3 - 5)² + (0 - 0)²) = √64 = 8

Point R (1,6) and Point Q (-1,2): distance = √((-1 - 1)² + (2 - 6)²) = √20 = 4.47

Point P (-1,0) and Point O (0,9): distance = √((0 - (-1))² + (9 - 0)²) = √82 = 9.06

Therefore, the only sets of points that have a distance of exactly 4 units between them are Point X and Point W, and Point T and Point S.

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The runner's average rate of change from 60 to 120 minutes is 0.093 miles per minute based on the given information.

Given that,

In the table, it shows the distance a runner has traveled.

The time (in minutes) and distance (in miles) values are as follows:

0 minutes: 0 miles

30 minutes: 3.48 miles

60 minutes: 6.42 miles

90 minutes: 9.18 miles

120 minutes: 12.0 miles

To find the runner's average rate of change from 60 to 120 minutes,

Use the formula:

The average rate of change = (Change in the distance) / (Change in time)

From the table,

At 60 minutes:

The runner has traveled 6.42 miles,

And at 120 minutes:

The runner has traveled 12.0 miles.

So, the change in distance is 12.0 miles - 6.42 miles = 5.58 miles.

The change in time is 120 minutes - 60 minutes = 60 minutes.

Now, we can calculate the average rate of change:

The average rate of change = 5.58 miles / 60 minutes

The average rate of change = 0.093 miles per minute.

Hence,

The runner's average rate of change from 60 to 120 minutes is 0.093 miles per minute.

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Answer:

The percent decrease is 36%.

Step-by-step explanation:

The value of the variable x is 2 in the given expression [tex]3\sqrt{x} - \sqrt{14x - 10} = 0[/tex].

Square rοοt οf a number is a value, which οn multiplicatiοn by itself, gives the οriginal number. The square rοοt is an inverse methοd οf squaring a number. Hence, squares and square rοοts are related cοncepts.

Suppοse x is the square rοοt οf y, then it is represented as x=√y, οr we can express the same equatiοn as x² = y. Here, ‘√’ is the radical symbοl used tο represent the rοοt οf numbers. The pοsitive number, when multiplied by itself, represents the square οf the number. The square rοοt οf the square οf a pοsitive number gives the οriginal number.

Find the value of x in the expression given below.

[tex]3\sqrt{x} - \sqrt{14x - 10} = 0[/tex]

Square both sides

[tex](3\sqrt{x} - \sqrt{14x - 10} )^2= 0^2[/tex]

Use (a - b)²

9x + 14x - 10 - 2([tex]3\sqrt{x} \times \sqrt{14x - 10}[/tex]) = 0

9x + 14x - 10 - 2([tex]3\sqrt{x} \times \sqrt{14x - 10}[/tex]) = 0

Use distributive property

23x - 10 - 43x + 50 = 0

-20x + 40 = 0

40 = 20x

x = 40/20

x = 2

Thus, the value of the variable x is 2 in the given expression [tex]3\sqrt{x} - \sqrt{14x - 10} = 0[/tex].

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The volume of the larger cylinder as 70 m³.

What is a cylinder, simply defined?

With two parallel bases joined by a curving surface, a cylinder is a three-dimensional solid. The bases are typically shaped like circles. The cylinder's height ("h") and radius ("r") are used to represent the perpendicular distance between the bases.

The scale factor of the cylinders = 5:2

The scaling factor indicates how much a figure has increased or decreased from its initial value.

The volume of the smaller cylinder = 28 m³

We are asked to find the volume of the larger cylinder.

let the volume of the larger cylinder be x.

x / 28 = 5/2

x  =(5/2) * 28 = 70

Therefore using the scale factor, we found the volume of the larger cylinder as 70 m³.

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Answer:

12

Step-by-step explanation:

The correct option is B an algorithm. If the wrong password is put in more than five times, Owen's computer will lock for an hour. To prevent this, he is building a programme. An illustration of a "algorithm" is this collection of instructions.

An algorithm is a process used to carry out a computation or solve a problem. In either hardware-based or software-based routines, algorithms operate as a detailed sequence of instructions that carry out prescribed operations sequentially.

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50 possible combinations

The probability that both of the rolls are made by the same participant is 25/99, since there are 25 participants and the second roll must be chosen from the remaining 99 rolls. The probability that both rolls have the same protein is 4/99, since there are 4 types of proteins and the second roll must be chosen from the remaining 99 rolls. Lastly, the probability that one roll has white rice and one roll has brown rice is 50/99, since there are 50 possible combinations of white and brown rice and the second roll must be chosen from the remaining 99 rolls.

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The required correct option out of given is B) This scale factor results in an enlarged image is correct.

Finding the dilation's center point is the first step in determining the scale factor. Next, we measure the distances between the center point and various points on the preimage and the image. According to Math Bits Notebook, the scale factor is equal to the ratio of these lengths.

When the scale factor of a dilation is greater than 1, the image is an enlargement.

In this case, the scale factor is 8:7, which means that every length in the image is 8/7 times larger than the corresponding length in the original figure. Therefore, the image is larger than the original figure.

So; B) This scale factor results in an enlarged image is correct.

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Answer: See attached.

Step-by-step explanation:

     First, we will find the unit rate by dividing.

312 miles / 8 gallons = 39 miles per gallon

     Then, we can fill out the other values using this unit rate.

39 miles / 39 miles per gallon = 1 gallon

3 gallons * 39 miles per gallon = 117 miles

       Lastly, we will use these two equivalent ratios we found above to fill out the table and then plot the points. See attached.

The correct option is B, The area of the figure will be the sum of the area of the rectangle and half the area of the circle is 220 cm².

The rectangle has a length of 22 cm and a width of 10 cm, so its area is:

A_rectangle = length × width = 22 cm × 10 cm = 220 cm²

A rectangle is a four-sided polygon with opposite sides parallel and equal in length. The area of a rectangle is the measure of the space that it occupies and is given by the product of its length and width.

To find the area of a rectangle, one needs to multiply the length of the rectangle by its width. For example, if the length of the rectangle is 5 meters and its width is 3 meters, the area would be 15 square meters. This is because 5 multiplied by 3 is equal to 15. In general, the formula for the area of a rectangle is A = lw, where A is the area, l is the length, and w is the width. The units of the area will depend on the units of the length and width.

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Complete Question: -

The drawing is composed of a rectangle and a semicircle Find the area of the figure to the nearest unit:

Not drawn to scale.

a. 41 cm^2

b. 220 cm^2

c. 410 cm^2

d. 820 cm2

To express (14, 10, 14) as a linear combination of u = (3, 1, 7), v = (1, -1, 6), and w = (7, 5, 5), we need to find values for a, b, and c such that:

(14, 10, 14) = a(3, 1, 7) + b(1, -1, 6) + c(7, 5, 5)

This can be written as a system of equations:

14 = 3a + b + 7c

10 = a - b + 5c

14 = 7a + 6b + 5c

We can solve this system of equations using any method we prefer, such as substitution or elimination. One possible solution is:

a = 1

b = 2

c = 1

Therefore, the linear combination of u, v, and w that gives us (14, 10, 14) is:

(14, 10, 14) = 1(3, 1, 7) + 2(1, -1, 6) + 1(7, 5, 5)

= (3, 1, 7) + (2, -2, 12) + (7, 5, 5)

= (14, 10, 14)

So, the linear combination is 1u + 2v + 1w.

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Answer:

Step-by-step explanation:

open the icon

The prοbability οf the events is:

P(A and B) = 7/100

P(A | B) = 16/25

P(A | B) = 0.495

P(A and B) = 0.175

The likelihοοd οr chance οf an event οccurring is measured by prοbability. The prοbability is expressed as a number between zerο (0) and οne (1), with zerο (0) shοwing that the event is impοssible, and οne (1) shοwing that the event is certain tο οccur. A prοbability οf 0.5 indicates that the event is equally likely tο οccur οr nοt οccur.

By dividing the number οf favοrable οutcοmes by the entire number οf pοssible pοssibilities, the prοbability οf an event is determined.

Fοr example, if we tοss a fair cοin, the prοbability οf getting heads is 1/2, because there is οne favοrable οutcοme (heads) οut οf twο pοssible οutcοmes (heads οr tails).

Therefοre, All events are dependent, sο the prοbability οf each event is:

P(A and B) = 7/100

P(A | B) = 16/25

P(A | B) = 0.495

P(A and B) = 0.175

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To solve the system of linear inequalities, we need to graph each inequality and find the region that satisfies both inequalities.

Here are the steps:
1. Graph the first inequality: -2x-y1. We can rearrange the equation to get y>-2x-1. This means that the region above the line y=-2x-1 is the solution to the first inequality.
2. Graph the second inequality: 3x+y<6. We can rearrange the equation to get y<-3x+6. This means that the region below the line y=-3x+6 is the solution to the second inequality.
3. The solution to the system of inequalities is the region that satisfies both inequalities. This is the region above the line y=-2x-1 and below the line y=-3x+6.
4. To find the coordinates of the vertices of the solution region, we can find the intersection point of the two lines. Setting the two equations equal to each other, we get: -2x-1=-3x+6. Solving for x, we get x=7. Substituting x=7 into one of the equations, we get y=-3(7)+6=-15. So the intersection point is (7,-15).
5. The solution region is the triangular region with vertices at (7,-15), (0,-1), and (2,0).

Therefore, the solution to the system of linear inequalities is the region with vertices at (7,-15), (0,-1), and (2,0).

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It will take Sean 3.75 hours to plant seeds in 0.5 square yards of the garden. Answer: 3.75 hours.

Multiplication is a basic arithmetic operation that involves combining two or more numbers to get a product. It is often represented using the multiplication symbol "*", and the numbers involved are referred to as factors. The result of a multiplication operation is called the product.

We can start by using the ratio of square yards to time as a proportion:

0.2 sq yd / 1.5 hrs = 0.5 sq yd / x hrs

Cross-multiplying, we get:

0.2 sq yd * x hrs = 1.5 hrs * 0.5 sq yd

Simplifying, we get:

x = (1.5 hrs * 0.5 sq yd) / 0.2 sq yd

x = 3.75 hrs

Therefore, it will take Sean 3.75 hours to plant seeds in 0.5 square yards of the garden. Answer: 3.75 hours.

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a) The derivative f'(x) is 12x².

b) The amount of f'(1) is 12.

c) The equation of the tangent line is 12x - 7.

a) To find the derivative of f(x), we can use the power rule for each term. The power rule states that for any function f(x) = xⁿ, the derivative f'(x) = nx^(n-1). So for f(x) = 4x³ + 1, Thus, the derivative f'(x) = 12x².

b) To compute the derivative of f at x=1, we simply plug in x=1 into the derivative equation we found in part a. So f'(1) = 12(1)² = 12.

c) To determine the equation of the tangent line to the curve y=f(x) at the point (1,5), we use the point-slope form of a line, which is y-y1 = m(x-x1), where m is the slope of the line and (x1,y1) is the point the line passes through. The slope of the tangent line is the derivative of f at x=1, which we found in part b to be 12. So the equation of the tangent line is y-5 = 12(x-1), or y = 12x - 7.

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Answer:

There are 875 seats. The stadium got $168,000 from the tickets.

Step-by-step explanation:

So there is more than one question so let's break them down one by one

1). How many seats are there in each section? So the stadium is divided into 12 sections and 8 are VIP boxes so 12 - 8 = 4 so there are 4 non-VIP sections. Now to find out how many seats are in each section we need to divide the number of seats by the number of sections which would be 10,500 ÷ 12 = 875.

2). If all the seats in the premium section are sold out for the game how many will the stadium get from those ticket sales? So first we need to find out how many seats there are in the premium section, we already know that there are 4 premium sections and we also know that there are 875 seats in each section. Multiply the number of sections by the number of seats 4 x 875 = 3,500 now multiply the number of seats by the cost of the seat 3,500 x $48 = $168,000

So the answers are $168,000 and 875 seats

Hope this helped :)

The area of the arrow shaped polygon in square units is: 80 square units

For triangle BCD:

b = Base = 4 - 2 = 2 units

h = Height = 4 units

Area = [tex]\frac{1}{2}[/tex]bh

Area = [tex]\frac{1}{2}[/tex] × 2 × 4

Area = 8 square units

For triangle DEF

b = Base = 2 - (-1) = 3 units

h = Height = 4 units

Area = [tex]\frac{1}{2}[/tex] × 3 ×  4

Area = 6 square units

For trapezoid ABFG:

a = 4 - (-1) = 5.5 units

b = 4 - (-8) = 12 units

h = 8 units

Area =  [tex]\frac{1}{2}[/tex](a + b)h

Area =  [tex]\frac{1}{2}[/tex](5.5 + 12)8

Area = 70 square units

Total area = 8 + 6 + 70

Total Area = 84 square units

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For 6 cities, the number of roads would be 9. This is because each city must be connected to every other city, and there are 5 pairs of cities that need to be connected, which means 5 roads.

Roads are a form of transportation infrastructure consisting of a prepared surface for vehicles to travel on. They are typically made from asphalt or concrete and are designed to provide a safe and efficient means of travel.

Then, the remaining 4 cities need to be connected, which means 4 more roads for a total of 9.
For 7 cities, the number of roads would be 12. This is because each city must be connected to every other city, and there are 6 pairs of cities that need to be connected, which means 6 roads. Then, the remaining 5 cities need to be connected, which means 5 more roads for a total of 12.
For N cities, the number of roads would be equal to N(N-1)/2. This is because each city must be connected to every other city, and there are N(N-1)/2 pairs of cities that need to be connected. For example, if N = 10, then 10 × 9 = 90 pairs of cities need to be connected, which means 90 roads.

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The number of $0.49$ stamps Paula bought is s = (1274 - 49s / 21) + 6.

To solve the equation 0.49s + 0.21(s - 6) = 12.74 for s, we can multiply both sides by 100 to get:

49s + 21(s - 6) = 1274

Now, we can move the 21(s - 6) term to the left side of the equation and the 49s term to the right side of the equation. To do this, we will subtract the 49s from both sides of the equation, which yields:

21(s - 6) = 1274 - 49s

Next, we can divide both sides of the equation by 21 to isolate the (s - 6) term, resulting in:

(s - 6) = 1274 - 49s / 21

Finally, we can add 6 to both sides of the equation to solve for the number of $0.49$ stamps that Paula bought:

s = (1274 - 49s / 21) + 6

Therefore, the number of $0.49$ stamps Paula bought is s = (1274 - 49s / 21) + 6.

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It will take 13 years for the account value to reach $10,800.

Compound interest means addition of interest to the principal sum of a loan or deposit which means interest on principal plus interest.

We will use the formula for compound interest [tex]A = P(1 + r/n)^{nt}[/tex]

Given that

A = Final account value ($10,800)

P = Principal amount ($4,500)

r = Annual interest rate (7% or 0.07)

n = 1

t = ?

$10,800 = $4,500(1 + 0.07/1)^(1*t)

Dividing by $4,500:

2.4 = (1.07)^t

Taking the logarithm of sides:

log(2.4) = log((1.07)^t)

Using logarithmic identity log(a^b) = b*log(a):

log(2.4) = t*log(1.07)

Dividing by log(1.07):

t = log(2.4) / log(1.07)

t = 12.9394949072

t = 13.

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r ≈ 12 meters is your answer :)

Anyway, it is from this formula: V=πr^ 2   h/ 3

it is actually ≈ 11.99696m

By answering the above question, we may state that Once everyone on unitary method the phone tree has been contacted and advised of the schedule change, the procedure may be repeated.

The unit technique is a strategy for addressing issues that entails first figuring out the value of a single unit, then multiplying that value to figure out the needed value. Simply expressed, the unit technique is used to extract a single unit value from a multiple that is provided.

First, the coordinator phones five persons to let them know that the school's schedule is changing.

Step 2: Those 5 people then each phone 5 more people, resulting in a total of 25 people receiving the news.

Step 3: The 25 call 5 additional individuals each, reaching a total of 125 people.

Step 4: The 125 call 5 additional individuals apiece, reaching a grand total of 625 persons.

Step 5: A total of 3,125 individuals have been told after the 625 phone 5 further persons each.

Once everyone on the phone tree has been contacted and advised of the schedule change, the procedure may be repeated.

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the values of a that satisfy the inequality |a|≤5 are those that satisfy both a ≤ 5 and a ≥ -5. This is why the expression equivalent to |a|≤5 is a ≥ –5 and a ≤ 5.

In mathematics, an expression is a combination of numbers, variables, and operators that can be evaluated to produce a value. It can be as simple as a single number or variable, or as complex as a long sequence of operations involving multiple variables and operators.

The expression |a|≤5 means that the absolute value of a is less than or equal to 5. The absolute value of a number is its distance from zero on the number line, regardless of whether the number is positive or negative.

So, if |a|≤5, then a must be between -5 and 5, including both.

To see why this is true, consider two cases:

If a is greater than or equal to zero, then |a| = a, and so we have a ≤ 5.

If a is less than zero, then |a| = -a, and so we have -a ≤ 5, which is equivalent to a ≥ -5.

Therefore, the values of a that satisfy the inequality |a|≤5 are those that satisfy both a ≤ 5 and a ≥ -5. This is why the expression equivalent to |a|≤5 is a ≥ –5 and a ≤ 5.

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Answer:

the values of a that satisfy the inequality |a|≤5 are those that satisfy both a ≤ 5 and a ≥ -5. This is why the expression equivalent to |a|≤5 is a ≥ –5 and a ≤ 5.

Step-by-step explanation:

What is expression ?

In mathematics, an expression is a combination of numbers, variables, and operators that can be evaluated to produce a value. It can be as simple as a single number or variable, or as complex as a long sequence of operations involving multiple variables and operators.

According to the given conditions :

The expression |a|≤5 means that the absolute value of a is less than or equal to 5. The absolute value of a number is its distance from zero on the number line, regardless of whether the number is positive or negative.

So, if |a|≤5, then a must be between -5 and 5, including both.

To see why this is true, consider two cases:

If a is greater than or equal to zero, then |a| = a, and so we have a ≤ 5.

If a is less than zero, then |a| = -a, and so we have -a ≤ 5, which is equivalent to a ≥ -5.

Therefore, the values of a that satisfy the inequality |a|≤5 are those that satisfy both a ≤ 5 and a ≥ -5. This is why the expression equivalent to |a|≤5 is a ≥ –5 and a ≤ 5.

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The number of people who had turkey on their sandwiches is 475.

option A.

Let's define the following;

A be the number of people who chose turkey,

B be the number of people who chose ham,

C be the number of people who chose both, and

D be the number of people who chose neither.

From the problem statement, we know that:

D = 75 (75 attendees chose neither)

C = 250 (250 attendees chose both)

B = 450 (450 attendees chose ham)

We want to find A (the number of attendees who chose turkey).

We can start by using the formula for the number of elements in the union of two sets:        

|A U B| = |A| + |B| - |A n B|

where;

Applying this formula to the problem, we get:

750 - D = A + B - C

750 - 75 = A + 450 - 250

675 = A + 200

A = 475

Therefore, 475 attendees chose turkey on their sandwiches. The answer is (A) 475.

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