A homeowner uses four wooden beams to build the perimeter of a rectangular deck. Each wooden beam has an equal length of x feet. He uses two beams for two sides of the deck, and he cuts off 4 feet from each of the other two beams for the remaining two sides of the deck, as shown by the diagram below.

Question

The total area of the deck, in square feet, is a function of the distance x, in feet.

What is the function, A(x), that represents the total area of the deck, in square feet?



(1 point)
Responses

A(x)=4x2−16x
cap A times x is equal to 4 x squared minus 16 x

A(x)=x2−4
cap A times x is equal to x squared minus 4

A(x)=2x−4
cap A times x is equal to 2 x minus 4

A(x)=x2−4x
cap A times x is equal to x squared minus 4 x

Answer 1

To find the function \( A(x) \) that represents the total area of the rectangular deck, we can start by outlining the dimensions of the deck based on the beams used.

1. Two sides of the rectangle are built using beams of length \( x \). Therefore, the length of these two sides is \( x \).

2. The other two sides are made from beams that have been cut down by 4 feet. Since each of these beams originally was \( x \) feet long, after cutting off 4 feet, the length of these two sides becomes \( x - 4 \).

Now, we have the dimensions of the deck:

• Length = \( x \)
• Width = \( x - 4 \)

The area \( A \) of a rectangle is given by the formula:

\[ A = \text{Length} \times \text{Width} \]

Substituting the lengths for Length and Width:

\[ A(x) = x \cdot (x - 4) \]

Now, simplify this expression:

\[ A(x) = x^2 - 4x \]

Therefore, the function that represents the total area of the deck, in square feet, is:

\[ \boxed{A(x) = x^2 - 4x} \]