Question

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.



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)=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

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

A(x)=x2−4

Answer 1

To find the function that represents the total area \( A(x) \) of the deck, let's analyze the setup based on the lengths of the beams.

1. **Lengths of the beams:**
- Two beams are used for two sides without cutting, giving a length of \( x \) each.
- The other two beams, which are cut by 4 feet, will have a length of \( x - 4 \) each.

2. **Dimensions of the rectangular deck:**
- Let the length of the deck be \( x \) (using the two uncut beams).
- Let the width of the deck be \( x - 4 \) (using the two cut beams).

3. **Area of the rectangular deck:**
- The area \( A \) of a rectangle is given by the formula:
\[
A = \text{length} \times \text{width}
\]
- Substituting the values we have:
\[
A(x) = x \cdot (x - 4)
\]
- Expanding this:
\[
A(x) = x^2 - 4x
\]

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

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