Question

the cones are similar. find the volume of the smaller cone. round to the nearest tenth.

Explanation:

Step1: Recall the ratio of volumes of similar solids

For two similar solids, if the ratio of their corresponding linear - dimensions (such as radii) is \(a:b\), the ratio of their volumes is \(a^{3}:b^{3}\). Let the radius of the smaller cone be \(r = 2\mathrm{cm}\) and the radius of the larger cone be \(R = 5\mathrm{cm}\). The ratio of the radii is \(\frac{r}{R}=\frac{2}{5}\), so the ratio of the volumes of the smaller cone to the larger cone is \((\frac{r}{R})^{3}=(\frac{2}{5})^{3}=\frac{8}{125}\).

Step2: Set up a proportion to find the volume of the smaller cone

Let \(V_{s}\) be the volume of the smaller cone and \(V_{l}=250\mathrm{cm}^{3}\) be the volume of the larger cone. We have the proportion \(\frac{V_{s}}{V_{l}}=\frac{8}{125}\). Substituting \(V_{l} = 250\mathrm{cm}^{3}\) into the proportion, we get \(V_{s}=\frac{8}{125}\times V_{l}\).

Step3: Calculate the volume of the smaller cone

\(V_{s}=\frac{8}{125}\times250\)
\[

$$\begin{align*} V_{s}&=\frac{8\times250}{125}\\ &=16\mathrm{cm}^{3} \end{align*}$$

\]

Answer:

\(16.0\mathrm{cm}^{3}\)