Answered

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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?

\begin{tabular}{|l|l|}
\hline
Volume of Square Pyramid & Volume of Cylinder \\
\hline
[tex]$V=\frac{1}{3} B(h)$[/tex] & [tex]$V=\pi r^2 h$[/tex] \\
\hline
[tex]$V=\frac{1}{3}(9.7^2)(9)$[/tex] & [tex]$V=\pi\left(5.47^2\right)(3)$[/tex] \\
\hline
[tex]$V=\frac{1}{3}(846.81)$[/tex] & [tex]$V=\pi(29.9209)(3)$[/tex] \\
\hline
[tex]$V=282 \, \text{in}^3$[/tex] & [tex]$V=\pi(89.7627)$[/tex] \\
\hline
& [tex]$V=282 \, \text{in}^3$[/tex] \\
\hline
\end{tabular}

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

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

Sagot :

GinnyAnsweridn
To determine if Jude's calculations are correct, let's examine the steps for calculating the volume of the square pyramid and the cylinder, and then compare the results he obtained by rounding to the nearest whole number.

### Volume of the Square Pyramid:

1. Formula: The volume [tex]\( V \)[/tex] of a square pyramid is calculated using the formula:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]

2. Base Area Calculation: The base of the square pyramid is a square with an edge length of 9.7 inches. The area of the base [tex]\( B \)[/tex] is:
[tex]\[ B = 9.7^2 = 94.09 \, \text{square inches} \][/tex]

3. Height: The height of the pyramid is 9 inches.

4. Volume Calculation: Substitute the base area and the height into the volume formula:
[tex]\[ V = \frac{1}{3} \times 94.09 \times 9 = \frac{1}{3} \times 846.81 = 282.27 \, \text{cubic inches} \][/tex]

5. Rounding: When rounded to the nearest whole number, the volume of the square pyramid is:
[tex]\[ \text{Rounded Volume} = 282 \, \text{cubic inches} \][/tex]

### Volume of the Cylinder:

1. Formula: The volume [tex]\( V \)[/tex] of a cylinder is calculated using the formula:
[tex]\[ V = \pi \times \text{Radius}^2 \times \text{Height} \][/tex]

2. Radius: The radius of the cylinder is 5.47 inches.

3. Height: The height of the cylinder is 3 inches.

4. Volume Calculation: Substitute the radius and the height into the volume formula, using [tex]\( \pi \approx 3.14159 \)[/tex]:
[tex]\[ V = \pi \times 5.47^2 \times 3 = \pi \times 29.9209 \times 3 = \pi \times 89.7627 = 281.99783888638444 \, \text{cubic inches} \][/tex]

5. Rounding: When rounded to the nearest whole number, the volume of the cylinder is:
[tex]\[ \text{Rounded Volume} = 282 \, \text{cubic inches} \][/tex]

### Conclusion:

When both volumes are calculated and rounded to the nearest whole number, they come out to 282 cubic inches. Thus, Jude's calculation shows that the volume of the square pyramid is equal to the volume of the cylinder when rounded to the nearest whole number. Therefore, Jude's rounded volumes are indeed correct.

Final Answer: Yes, his calculations are correct, and the volumes for the figures are equal when rounded to the nearest whole number.