To determine whether Jude's calculations are correct, we will calculate the volumes of both the square pyramid and the cylinder step by step.
Volume of the Square Pyramid:
The volume \( V \) of a square pyramid can be calculated using the formula:
\[ V = \frac{1}{3} B h \]
where \( B \) is the area of the base and \( h \) is the height. The base is a square with edges of 9.7 inches, so the area \( B \) can be calculated as:
\[ B = (9.7)^2 = 94.09 \text{ in}^2 \]
Now, using the height \( h = 9 \text{ in} \):
\[ V = \frac{1}{3} (94.09) (9) = \frac{1}{3} (846.81) \approx 282.27 \text{ in}^3 \]
Rounded to the nearest whole number, the volume of the square pyramid is 282 in³.
Volume of the Cylinder:
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height. Given a radius of 5.47 inches and height of 3 inches, we can compute:
First, calculate \( r^2 \):
\[ r^2 = (5.47)^2 \approx 29.9209 \text{ in}^2 \]
Now, substituting this into the volume formula:
\[ V \approx \pi(29.9209)(3) \approx 3.14159 \times 89.7627 \approx 282.61 \text{ in}^3 \]
Rounded to the nearest whole number, the volume of the cylinder is 283 in³.
Comparison of Volumes:
- Volume of the square pyramid: 282 in³
- Volume of the cylinder: 283 in³
Conclusion:
Since the volumes of the square pyramid and the cylinder are not equal (282 in³ ≠ 283 in³), Jude made a mistake.
The correct answer is: No, he made a mistake in solving for the volume of the cylinder.
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