To determine the area of the entire quilt as a function of \( x \), we will first find the dimensions of the quilt based on the given information about the first fabric scrap.
- The width of the first scrap is \( 2x \) feet.
- The length of the first scrap is \( 5x \) feet.
Now, we need to calculate the dimensions of the entire quilt. According to the problem:
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The quilt will be 5 feet wider than the first fabric scrap, so the width of the quilt is: \[ \text{Width of quilt} = 2x + 5 \]
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The quilt will be 3 feet longer than the first fabric scrap, so the length of the quilt is: \[ \text{Length of quilt} = 5x + 3 \]
Next, we can calculate the area of the quilt by multiplying its width and length: \[ \text{Area of quilt} = (\text{Width of quilt}) \times (\text{Length of quilt}) = (2x + 5)(5x + 3) \]
Now, we will expand this expression: \[ (2x + 5)(5x + 3) = 2x(5x) + 2x(3) + 5(5x) + 5(3) \] \[ = 10x^2 + 6x + 25x + 15 \] \[ = 10x^2 + 31x + 15 \]
Thus, the function \( f(x) \) that represents the area of the quilt in square feet is: \[ f(x) = 10x^2 + 31x + 15 \]
The correct answer is: C. \( f(x) = 10x^2 + 31x + 15 \)
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