Question

Two similar cones have radii of 6 and 1, respectively. What is the ratio of their volumes?

Answer 1

Correct answer is D

Explanation:

Two similar cones have radii of
`R=6` units and
`r=1` unit. Then the coefficient of similarity is

`k=(R)/(r)=(6)/(1)=6.`

The similar shapes have all linear lengths proportional. If H is the height of the larger cone and h is the height of the smaller cone, then

`(H)/(h)=k,\\ \\(H)/(h)=6\Rightarrow H=6h.`

Use formula for the volume of the cone:

`V_(cone)=(1)/(3)\cdot \pi r^2\cdot h.`

Then

`\frac{V_{\text{large cone}}}{V_{\text{smal cone}}}=((1)/(3)\cdot \pi R^2\cdot H)/((1)/(3)\cdot \pi r^2\cdot h)=((6)^2\cdot 6h)/(1^2\cdot h)=(36\cdot 6)/(1)=(216)/(1).`