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)=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
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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)=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
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page 10 of 10
Answers
GPT-4o mini
To determine the function \( A(x) \) that represents the total area of the deck, we start with understanding the dimensions based on the wooden beams.
1. **Sides of the Deck:**
- The two sides of the deck that use the full wooden beams will each have a length of \( x \) feet.
- The other two sides will be made from beams that have been cut off by 4 feet, resulting in a length of \( x - 4 \) feet each.
2. **Area of the Rectangle:**
- The area \( A \) of a rectangle is given by the formula:
\[
A = \text{length} \times \text{width}
\]
- In this case, the length is \( x \) and the width is \( x - 4 \).
3. **Function Representation:**
- Substituting the lengths into the area formula gives:
\[
A(x) = x \cdot (x - 4)
\]
- Expanding the equation:
\[
A(x) = x^2 - 4x
\]
Thus, the function \( A(x) \) that represents the total area of the deck, in square feet, is:
\[
\boxed{A(x) = x^2 - 4x}
\]
This corresponds to the choice:
- **A(x) = x^2 - 4x**
1. **Sides of the Deck:**
- The two sides of the deck that use the full wooden beams will each have a length of \( x \) feet.
- The other two sides will be made from beams that have been cut off by 4 feet, resulting in a length of \( x - 4 \) feet each.
2. **Area of the Rectangle:**
- The area \( A \) of a rectangle is given by the formula:
\[
A = \text{length} \times \text{width}
\]
- In this case, the length is \( x \) and the width is \( x - 4 \).
3. **Function Representation:**
- Substituting the lengths into the area formula gives:
\[
A(x) = x \cdot (x - 4)
\]
- Expanding the equation:
\[
A(x) = x^2 - 4x
\]
Thus, the function \( A(x) \) that represents the total area of the deck, in square feet, is:
\[
\boxed{A(x) = x^2 - 4x}
\]
This corresponds to the choice:
- **A(x) = x^2 - 4x**