Question

the figure below shows two similar cones, one inside the other, with their bases on the same plane. the volume of the smaller cone is π cu in. what is the volume, v, of the larger cone? 2 of 5 question v = 32π cu in v = 24π cu in v = 18π cu in v = 8π cu in

Explanation:

Step1: Determine the scale factor of heights

The height of the smaller cone is \(3\) in and the height of the larger cone is \(3 + 3=6\) in. The scale factor of heights \(k=\frac{h_{large}}{h_{small}}=\frac{6}{3} = 2\).

Step2: Use the volume - scale factor relationship for similar solids

For similar solids, if the scale factor of corresponding linear dimensions (such as height) is \(k\), the scale factor of volumes is \(k^{3}\). Let \(V_{small}=\pi\) and \(V_{large}\) be the volumes of the smaller and larger cones respectively. We know that \(\frac{V_{large}}{V_{small}}=k^{3}\).
Since \(k = 2\), then \(V_{large}=V_{small}\times k^{3}\).
Substitute \(V_{small}=\pi\) and \(k = 2\) into the formula: \(V_{large}=\pi\times2^{3}\).

$$V_{large}=8\pi$$

Answer:

\(V = 8\pi\) cu in (the fourth option)