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?
A) A(x) = 4x^2 − 16x
B) A(x) = x^2 − 4x
C) A(x) = x^2 − 4
D) A(x) = 2x − 4

Answer 1

To find the function \( A(x) \) that represents the total area of the deck, we can analyze the problem using the information given.

1. **Length of the sides**:
- For the two sides using full beams: The length of these sides is \( x \) feet each.
- For the other two sides, which are cut down by 4 feet: The length of these sides is \( x - 4 \) feet each.

2. **Calculating the area**:
- The area \( A \) of a rectangle is given by the formula:
\[
A = \text{length} \times \text{width}
\]
- In our case, the length can be taken as \( x \) (the longer sides) and the width as \( x - 4 \) (the shorter sides).
- Thus, the area \( A(x) \) can be expressed as:
\[
A(x) = x \times (x - 4)
\]

3. **Expanding the area expression**:
- Now we will expand the area expression:
\[
A(x) = x^2 - 4x
\]

4. **Final result**:
- Therefore, the function that represents the total area of the deck, in square feet, is:
\[
A(x) = x^2 - 4x
\]

From the provided options, this matches choice **B**.

So the correct answer is:
**B) A(x) = x^2 − 4x**