Jude says that the volume of a square pyramid with base edges of 9.7 in and a height of 9 in is equal to the volume of a cylinder with a radius of 5.47 in and a height of 3 in. Jude rounded his answers to the nearest whole number. Examine Jude's calculations. Is he correct? (4 points)

Question

Volume of Square Pyramid Volume of Cylinder
V = one third B(h) V = πr2h
V = one third(94.09)(9) V = π(5.472)(3)
V = one third(846.81) V = π(29.9209)(3)
V = 282 in3 V = π(89.7627)

V ≈ 282 in3

Yes, his calculations are correct and the volumes for figures are equal.

No, he made a mistake in solving for the volume of the cylinder.

Yes, but he made a mistake in solving for the volume of the square pyramid.

No, he made a mistake in solving for the volume of both figures.

Answer 1

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:

$$V=\frac{1}{3}Bh$$

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.

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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.

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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.