Question

find the volume of a cone that has a radius of 8 inches and a slant height of 10 inches. round your answer to the nearest tenth of a cubic inch.
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Explanation:

Step1: Find the height of the cone

We know that for a cone, the slant height \( l \), radius \( r \), and height \( h \) satisfy the Pythagorean theorem \( l^{2}=r^{2}+h^{2} \). Given \( r = 8 \) inches and \( l=10 \) inches, we can solve for \( h \):
\( h=\sqrt{l^{2}-r^{2}}=\sqrt{10^{2}-8^{2}}=\sqrt{100 - 64}=\sqrt{36} = 6 \) inches.

Step2: Calculate the volume of the cone

The formula for the volume \( V \) of a cone is \( V=\frac{1}{3}\pi r^{2}h \). Substitute \( r = 8 \) inches and \( h = 6 \) inches into the formula:
\( V=\frac{1}{3}\times\pi\times8^{2}\times6 \)
First, calculate \( 8^{2}=64 \), then:
\( V=\frac{1}{3}\times\pi\times64\times6 \)
Simplify \( \frac{1}{3}\times6 = 2 \), so \( V = 2\times\pi\times64=128\pi \)
Using \( \pi\approx3.14159 \), we get \( V\approx128\times3.14159\approx402.12384 \)
Rounding to the nearest tenth, \( V\approx402.1 \) cubic inches.

Answer:

\( 402.1 \)