Question

these solids are similar. find the volume of the smaller solid. r = 3 m v = ? m³ r = 5 m v = 250 m³

Explanation:

Step1: Find the ratio of the radii

The ratio of the radii of the smaller cone to the larger cone is \(\frac{r_{1}}{r_{2}}=\frac{3}{5}\).

Step2: Use the volume ratio formula for similar solids

For similar solids, the ratio of their volumes is \((\frac{r_{1}}{r_{2}})^{3}\). Let \(V_{1}\) be the volume of the smaller cone and \(V_{2} = 250\space m^{3}\) be the volume of the larger cone. Then \(\frac{V_{1}}{V_{2}}=(\frac{3}{5})^{3}\).
Substitute \(V_{2} = 250\) into the formula: \(V_{1}=V_{2}\times(\frac{3}{5})^{3}\).
Calculate \((\frac{3}{5})^{3}=\frac{27}{125}\).
Then \(V_{1}=250\times\frac{27}{125}\).

Step3: Simplify the expression

\(250\times\frac{27}{125}=\frac{250\times27}{125}\).
Since \(250\div125 = 2\), then \(V_{1}=2\times27\).

Answer:

\(54\space m^{3}\)