To find the unit rate of area in square feet of the enlarged image per square foot of area of the original image, we first need to calculate the area of both the enlarged image and the original image.
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Calculate the area of the enlarged image: The dimensions of the enlarged image are given as:
- Length = 8 feet
- Width = 6 feet
The area of the enlarged image (A_enlarged) can be calculated as: \[ A_{\text{enlarged}} = \text{Length} \times \text{Width} = 8 , \text{feet} \times 6 , \text{feet} = 48 , \text{square feet} \]
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Determine the scaling factor: The scale is 1/4 foot to 3 feet. This can be interpreted as:
- The enlarged image is \( 3 / (1/4) \) times larger than the original, which can be simplified: \[ \text{Scaling factor} = 3 \div \frac{1}{4} = 3 \times 4 = 12 \]
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Calculate the area of the original image: Since the enlarged image is 12 times larger in every dimension, its area will be: \[ A_{\text{original}} = \frac{A_{\text{enlarged}}}{\text{Scaling factor}^2} = \frac{48 , \text{square feet}}{12^2} = \frac{48 , \text{square feet}}{144} = \frac{1}{3} , \text{square feet} \]
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Calculate the unit rate of area of the enlarged image per square foot of the original image: \[ \text{Unit rate} = \frac{A_{\text{enlarged}}}{A_{\text{original}}} = \frac{48 , \text{square feet}}{\frac{1}{3} , \text{square feet}} = 48 \times 3 = 144 , \text{square feet per square foot} \]
Thus, the unit rate of area in square feet of the enlarged image per square foot of area of the original image is A. 144 square feet per square foot.
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