On a website selling books, three paperback books and five hardcover books cost $80.10, and seven paperback books and four hardc

Solution:

Refer to the detailed explanation

Step-by-step process:

1 step: - provided

2 step: - Segments Addition Postulate

3 step: - Substitution Property

4 step: - Segments Addition Postulate

5 step: - Substitution Property

6 step: - provided

7 step: - Property of Substitution Equality

8 step: - Equality Subtraction Property

Answer:

The three errors are:

1) Incorrectly swapping the variables x and y.

2) Failure to use the ± symbol.

3) The domain is wrong; it should be x ≤ 0.

Step-by-step explanation:

Review the following steps.

Step 1:

The initial step is incorrect because the variables x and y were switched improperly; it should be:

Step 2:

Step 3:

Step 4: , for x ≥ 0

Step 4 contains an error as the ± sign is required. Additionally, the function's domain is inaccurately stated.

The radicand must be nonnegative, implying that x must satisfy x ≤ 0.

Answer:

Part 1) The equation is

Part 2) When x=40 m, the area of the schoolyard is A=2,800 m^2

Part 3) The valid domain consists of all real numbers exceeding zero and below 75 meters

Step-by-step explanation:

Part 1) Formulate an expression for A(x)

Let

x -----> the length of the rectangular school yard

y ---> the width of the rectangular school yard

It is known that

The perimeter for the fencing (taking the school wall as one side) is

thus

-----> this is equation A

The area of the rectangular school yard is

----> this is equation B

Substituting equation A into equation B yields

Change to function notation

Part 2) What is the area when x=40?

With x equal to 40 m

substitute the value into the expression from Part 1 to determine A

Part 3) What would be a suitable domain for A(x) in this scenario?

We understand that

A signifies the area of the rectangular school yard

x characterizes the length of the rectangular school yard

It follows that

This forms a vertical parabola opening downwards

The vertex indicates a maximum point

The x-coordinate of the vertex corresponds to the length that maximizes the area

The y-coordinate of the vertex denotes the maximum area

The vertex corresponds to (37.5, 2812.5)

Refer to the accompanying figure

Consequently,

The peak area achieved is 2,812.5 m^2

The x-intercepts are located at x=0 m and x=75 m

The domain for A is the range -----> (0, 75)

All real numbers greater than zero and less than 75 meters