Question

A solid is formed by cutting a conical section away a right circular cylinder. If the radius measures 6 in. and the altitude measures 8 in., what is the volume of the resulting solid?

Conic sections
ANSWERED
asked 2021-03-11
A solid is formed by cutting a conical section away a right circular cylinder. If the radius measures 6 in. and the altitude measures 8 in., what is the volume of the resulting solid?

Answers (1)

2021-03-12

Step 1
Find the volume of the resulting solid.
Step 2
Let the volume of the circular cylinder be \(V_{1}\ \text{and let the volume of the right circular cone be}\ V_{2}.\)
A solid formed by cutting a conical section away from a right circular cylinder.
Find the volume \(V_{1}.\)
\(\displaystyle{V}_{{1}}=\pi{r}^{2}{h}\)
\(\displaystyle=\pi{\left({6}\right)}^{2}{\left({8}\right)}\)
\(= 288\ \pi\)
Find the volume \(V_{2}.\)
\(\displaystyle{V}_{{2}}=\frac{1}{{3}}\pi{r}^{2}{h}\)
\(\displaystyle=\frac{1}{{3}}\pi{\left({6}^{2}{\left({8}\right)}\right.})\)
\(= 96\ \pi\)
Step 3
Find the volume of the resulting solid.
\(\displaystyle{V}={V}_{{1}}-{V}_{{2}}\)
\(\displaystyle={288}\pi-{96}\pi\)
\(= 192\ \pi\)
\(\displaystyle\approx{603}.{19}\in^{2}\)
Hence, the volume of the resulting solid \(\displaystyle{603.19}\in^{2}\)

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