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)=x2−4
cap A times x is equal to x squared 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)=2x−4
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)=x2−4
cap A times x is equal to x squared 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)=2x−4
Answers
GPT-4o mini
To determine the function \( A(x) \) that represents the total area of the rectangular deck, we need to first express the dimensions of the deck in terms of \( x \).
- Two beams of length \( x \) are used for the length of the deck, so one side of the deck (length) is \( x \) feet.
- For the other two sides of the deck, the beams are cut down by 4 feet. Therefore, each of these beams has a length of \( x - 4 \) feet. Hence, the other side of the deck (width) is \( x - 4 \) feet.
Now we can express the area \( A \) of the rectangle using the formula for the area of a rectangle, which is length times width:
\[
A(x) = \text{length} \times \text{width} = x \times (x - 4)
\]
Expanding this expression:
\[
A(x) = x(x - 4) = x^2 - 4x
\]
Thus, the function representing the total area of the deck in square feet is:
\[
A(x) = x^2 - 4x
\]
The correct response is:
**A(x) = x² - 4x**
- Two beams of length \( x \) are used for the length of the deck, so one side of the deck (length) is \( x \) feet.
- For the other two sides of the deck, the beams are cut down by 4 feet. Therefore, each of these beams has a length of \( x - 4 \) feet. Hence, the other side of the deck (width) is \( x - 4 \) feet.
Now we can express the area \( A \) of the rectangle using the formula for the area of a rectangle, which is length times width:
\[
A(x) = \text{length} \times \text{width} = x \times (x - 4)
\]
Expanding this expression:
\[
A(x) = x(x - 4) = x^2 - 4x
\]
Thus, the function representing the total area of the deck in square feet is:
\[
A(x) = x^2 - 4x
\]
The correct response is:
**A(x) = x² - 4x**