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.
Deck
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)
A(z) = 2x-4
Deck
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)
A(z) = 2x-4
Answers
GPT-4o mini
To find the area function \( A(x) \) for the rectangular deck described, we first need to determine the dimensions of the deck based on the provided lengths of the wooden beams.
1. The two long sides of the rectangle use the full length of the beams, which is \( x \) feet each. Therefore:
- The length of the deck \( L \) is \( x \).
2. The two short sides of the rectangle each have 4 feet cut off from their respective beams. Since each of these beams has a length of \( x \), the length of each of the short sides becomes:
- The width of the deck \( W \) is \( x - 4 \).
Now we can write the formula for the area \( A \) of a rectangle, which is given by the product of its length and width:
\[
A(x) = L \times W
\]
Substituting the expressions we found for \( L \) and \( W \):
\[
A(x) = x \times (x - 4)
\]
Expanding the expression:
\[
A(x) = x^2 - 4x
\]
Thus, the function that represents the total area of the deck in square feet as a function of \( x \) is:
\[
A(x) = x^2 - 4x
\]
1. The two long sides of the rectangle use the full length of the beams, which is \( x \) feet each. Therefore:
- The length of the deck \( L \) is \( x \).
2. The two short sides of the rectangle each have 4 feet cut off from their respective beams. Since each of these beams has a length of \( x \), the length of each of the short sides becomes:
- The width of the deck \( W \) is \( x - 4 \).
Now we can write the formula for the area \( A \) of a rectangle, which is given by the product of its length and width:
\[
A(x) = L \times W
\]
Substituting the expressions we found for \( L \) and \( W \):
\[
A(x) = x \times (x - 4)
\]
Expanding the expression:
\[
A(x) = x^2 - 4x
\]
Thus, the function that represents the total area of the deck in square feet as a function of \( x \) is:
\[
A(x) = x^2 - 4x
\]