To find the function that represents the area \( f(x) \) of the quilt, we first need to determine the dimensions of the quilt based on the information provided.
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The dimensions of the first scrap:
- Width = \( 2x \) feet
- Length = \( 5x \) feet
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The area of the first fabric scrap: \[ \text{Area of the first scrap} = \text{Width} \times \text{Length} = 2x \times 5x = 10x^2 \text{ square feet} \]
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The quilt will be 5 feet wider and 3 feet longer than the first scrap:
- Width of the quilt = Width of the scrap + 5 = \( 2x + 5 \) feet
- Length of the quilt = Length of the scrap + 3 = \( 5x + 3 \) feet
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Now, we calculate the area of the quilt: \[ \text{Area of the quilt} = \text{Width of the quilt} \times \text{Length of the quilt} \] \[ \text{Area of the quilt} = (2x + 5)(5x + 3) \]
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We now expand this product: \[ (2x + 5)(5x + 3) = 2x(5x) + 2x(3) + 5(5x) + 5(3) \] \[ = 10x^2 + 6x + 25x + 15 \] \[ = 10x^2 + 31x + 15 \]
Therefore, the function that gives the area of the quilt in square feet is: \[ f(x) = 10x^2 + 31x + 15 \]
The correct answer is C.
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