Question
Question 1 A) 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−4A(x)=2x−4cap A times x is equal to 2 x minus 4 A(x)=x2−4A(x)=x2−4cap A times x is equal to x squared minus 4 A(x)=x2−4xA(x)=x2−4xcap A times x is equal to x squared minus 4 x A(x)=4x2−16xA(x)=4x2−16xcap A times x is equal to 4 x squared minus 16 x Question 2 A)The length of a rectangle is 2 inches less than 3 times the width. Which equation describes the relationship between w, the width of the rectangle in inches, and A, the area in square inches?(1 point) Responses A=2w2−3wA=2w2−3wcap A is equal to 2 w squared minus 3 w A=3w2−2A=3w2−2cap A is equal to 3 w squared minus 2 A=2w2−3A=2w2−3cap A is equal to 2 w squared minus 3 A=3w2−2wA=3w2−2wcap A is equal to 3 w squared minus 2 w Question 3 A) The graph below shows the relationship between the side length in feet of a square piece of carpet and its price in dollars. If s is the side length and p is the price, which of these equations does the graph represent? (1 point) Responses p=10sp=10sp is equal to 10 s p=s2p=s2p is equal to s squared p=5sp=5sp is equal to 5 s p=2.5s2p=2.5s2p is equal to 2 point 5 s squared
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GPT-4o mini
### Question 1:
To find the function A(x) that represents the total area of the rectangular deck, we first need to identify the dimensions of the deck based on the information given.
- The length of the two longer sides of the rectangle is \( x \) feet (using two beams full length).
- The length of the two shorter sides, after cutting 4 feet from each beam, is \( x - 4 \) feet.
The area \( A \) of a rectangle is given by the formula:
\[ A = \text{length} \times \text{width} \]
Substituting in the values we found:
\[ A(x) = x \times (x - 4) \]
\[ A(x) = x^2 - 4x \]
Thus, the function that represents the area of the deck is:
**A(x) = x² - 4x**
### Question 2:
Given that the length \( L \) of a rectangle is 2 inches less than 3 times the width \( w \), we can express the length in terms of width:
\[ L = 3w - 2 \]
The area \( A \) of a rectangle is calculated as:
\[ A = L \times w \]
Substituting the expression for length:
\[ A = (3w - 2) \times w \]
\[ A = 3w^2 - 2w \]
Thus, the equation that describes the relationship between the width \( w \) and the area \( A \) is:
**A = 3w² - 2w**
### Question 3:
The question states there is a relationship between the side length \( s \) of a square piece of carpet and its price \( p \). In general, for a square, the area (which could be correlated with price in this context) can be given as:
\[ \text{Area} = s^2 \]
If we consider the price directly proportional to the area, we can express the price in terms of \( s \):
1. If \( p = k \cdot s^2 \) for some constant \( k \).
2. We can eliminate constants and focus on either a proportional relation or a specific k from the responses.
From the options provided, if the graph shows a straightforward relationship of price proportional to area, the simplest assumption is:
\[ p = s^2 \]
Thus, if the graph represents the relation of price \( p \) as equal to a multiple of the square of the side length \( s \) or directly proportional in relationships laid out, I would conclude:
**p = s²**
This implies it aligns with the idea of square area directly representing total price. If there are hints/contexts about the constant \( 10 \), \( 5 \), or \( 2.5 \), one would check against those details presented in the graph.
To find the function A(x) that represents the total area of the rectangular deck, we first need to identify the dimensions of the deck based on the information given.
- The length of the two longer sides of the rectangle is \( x \) feet (using two beams full length).
- The length of the two shorter sides, after cutting 4 feet from each beam, is \( x - 4 \) feet.
The area \( A \) of a rectangle is given by the formula:
\[ A = \text{length} \times \text{width} \]
Substituting in the values we found:
\[ A(x) = x \times (x - 4) \]
\[ A(x) = x^2 - 4x \]
Thus, the function that represents the area of the deck is:
**A(x) = x² - 4x**
### Question 2:
Given that the length \( L \) of a rectangle is 2 inches less than 3 times the width \( w \), we can express the length in terms of width:
\[ L = 3w - 2 \]
The area \( A \) of a rectangle is calculated as:
\[ A = L \times w \]
Substituting the expression for length:
\[ A = (3w - 2) \times w \]
\[ A = 3w^2 - 2w \]
Thus, the equation that describes the relationship between the width \( w \) and the area \( A \) is:
**A = 3w² - 2w**
### Question 3:
The question states there is a relationship between the side length \( s \) of a square piece of carpet and its price \( p \). In general, for a square, the area (which could be correlated with price in this context) can be given as:
\[ \text{Area} = s^2 \]
If we consider the price directly proportional to the area, we can express the price in terms of \( s \):
1. If \( p = k \cdot s^2 \) for some constant \( k \).
2. We can eliminate constants and focus on either a proportional relation or a specific k from the responses.
From the options provided, if the graph shows a straightforward relationship of price proportional to area, the simplest assumption is:
\[ p = s^2 \]
Thus, if the graph represents the relation of price \( p \) as equal to a multiple of the square of the side length \( s \) or directly proportional in relationships laid out, I would conclude:
**p = s²**
This implies it aligns with the idea of square area directly representing total price. If there are hints/contexts about the constant \( 10 \), \( 5 \), or \( 2.5 \), one would check against those details presented in the graph.
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