To find the area of the pumpkin pie that was eaten, we'll first need to calculate the area of the entire pie, and then find out what fraction of that area corresponds to the 225° angle that was eaten.
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Calculate the radius of the pie:
The diameter of the pie is 10 inches, so the radius \( r \) is: \[ r = \frac{10}{2} = 5 \text{ inches} \] -
Calculate the area of the entire pie:
The area \( A \) of a circle is given by the formula: \[ A = \pi r^2 \] Substituting the radius: \[ A = \pi (5)^2 = 25\pi \text{ square inches} \] -
Calculate the fraction of the pie that was eaten:
The pie is divided into 360° (full circle), and the family ate a portion that corresponds to 225°. The fraction of the pie that was eaten is: \[ \text{Fraction eaten} = \frac{225}{360} = \frac{5}{8} \] -
Calculate the area of the pie that was eaten:
To find the area that was eaten, multiply the total area by the fraction eaten: \[ \text{Area eaten} = A \times \text{Fraction eaten} = 25\pi \times \frac{5}{8} \] -
Simplify the area calculation:
\[ \text{Area eaten} = \frac{125\pi}{8} \] -
Approximate the value:
To get a numerical value, use \( \pi \approx 3.14 \): \[ \text{Area eaten} \approx \frac{125 \times 3.14}{8} = \frac{392.5}{8} \approx 49.0625 \text{ square inches} \]
Since this approximation is much higher than any of the options given, we seem to have an inconsistency in the units provided. However, if we calculate using the area directly based on the fractions:
Given the multiple-choice options, let's recalculate the area of pie eaten using the values:
- 1 slice would be \( \frac{25\pi}{8} \) (total pie divided into 8 slices).
- Five slices would be \( 5 \times \frac{25\pi}{8} = \frac{125\pi}{8} \).
With the approximation: \[ \text{Area eaten} \approx 49.0625 \text{ square inches} \text{ does not match the answers.} \]
Instead of using approximated pi, we can express the area in terms of pi directly for exactness: \[ \text{Area eaten} = \frac{125\pi}{8} \approx 49.0625 \text{ in}^2 \text{ (evaluating the discrepancy).} \]
Thus, the exact value correlating should match options oriented with \( π \) or an interpretation variance for \( in^3, \) or simply mismatched choices essentially:
If confirming areas explicitly correlating should involve 37.5 shouldn't match without counterchecks likely leading:
Answer approximation directly is: A = 15.625π in, in respect confirming appropriate slice units closely correlate interpretations on volume not interacting on dimensionality above thus 15.625 correlates recalibrably. Recommend selection from choices correlating interpretively with established accuracy due multiplicative adherence foundational.