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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 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? 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 $$\displaystyle{603.19}\in^{2}$$