Answered

Jude says that the volume of a square pyramid with base edges of 7 in and a height of 7 in is equal to the volume of a cylinder with a radius of 7 in and a height of 7 in. 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]$V=\frac{1}{3}(49)(7)$[/tex] & [tex]$V=\pi(7^2)(7)$[/tex] \\
\hline
[tex]$V=\frac{343}{3} \text{ in}^3$[/tex] & [tex]$V=343\pi \text{ in}^3$[/tex] \\
\hline
\end{tabular}

A. Yes, his calculations are correct and the volumes for the figures are equal.
B. No, he made a mistake in solving for the volume of the square pyramid.
C. Yes, but he made a mistake in solving for the volume of the cylinder.
D. No, he made a mistake in solving for the volume of both figures.



Answer :

GinnyAnswer
Let's examine both volumes step-by-step to determine if Jude's calculations are correct.

### Volume of the Square Pyramid:
The formula for the volume of a square pyramid is given by:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times \text{base area} \times \text{height} \][/tex]

1. Base Edge Length (a) = 7 inches
- The base area (B) of the square pyramid:
[tex]\[ \text{Base Area} = a^2 = 7^2 = 49 \text{ square inches} \][/tex]

2. Height (h) = 7 inches
- Now, calculate the volume of the pyramid:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times 49 \times 7 \][/tex]

3. Calculate the volume:
[tex]\[ V_{\text{pyramid}} = \frac{1}{3} \times 343 \][/tex]
[tex]\[ V_{\text{pyramid}} = 114.33 \approx 114 \text{ cubic inches} \][/tex] (Rounded to the nearest whole number)

### Volume of the Cylinder:
The formula for the volume of a cylinder is given by:
[tex]\[ V_{\text{cylinder}} = \pi \times r^2 \times h \][/tex]

1. Radius (r) = 7 inches
2. Height (h) = 7 inches
- Now, calculate the volume of the cylinder:
[tex]\[ V_{\text{cylinder}} = \pi \times 7^2 \times 7 \][/tex]
[tex]\[ V_{\text{cylinder}} = \pi \times 49 \times 7 \][/tex]
[tex]\[ V_{\text{cylinder}} = 343\pi \][/tex]

3. Using the value of [tex]\(\pi \approx 3.14\)[/tex] for practical calculations:
[tex]\[ V_{\text{cylinder}} ≈ 3.14 \times 343 \][/tex]
[tex]\[ V_{\text{cylinder}} ≈ 1078.22 \approx 1078 \text{ cubic inches} \][/tex] (Rounded to the nearest whole number)

### Comparison:
- Volume of the square pyramid [tex]\( \approx 114 \)[/tex] cubic inches.
- Volume of the cylinder [tex]\( \approx 1078 \)[/tex] cubic inches.

Conclusion:
Jude's statement that the volumes of both figures are equal is incorrect.

The correct conclusion based on the step-by-step calculations above is:
No, he made a mistake in solving for the volume of both figures.