Question

these solids are similar. find the surface area of the smaller solid. r = 5 cm sa = ? cm² r = 6 cm sa = 756 cm²

Explanation:

Step1: Find the ratio of the radii

The ratio of the radii of the smaller cylinder to the larger cylinder is \(\frac{r_{1}}{r_{2}}=\frac{5}{6}\).

Step2: Use the ratio of surface - areas formula

For similar solids, the ratio of their surface - areas \(SA\) is \((\frac{r_{1}}{r_{2}})^2\). Let \(SA_1\) be the surface - area of the smaller cylinder and \(SA_2 = 756\space cm^{2}\) be the surface - area of the larger cylinder. Then \(\frac{SA_{1}}{SA_{2}}=(\frac{r_{1}}{r_{2}})^2\).
Substitute \(r_{1} = 5\), \(r_{2}=6\) and \(SA_{2}=756\) into the formula: \(\frac{SA_{1}}{756}=(\frac{5}{6})^2=\frac{25}{36}\).

Step3: Solve for \(SA_1\)

Cross - multiply: \(SA_{1}=\frac{25\times756}{36}\).
First, calculate \(25\times756 = 18900\). Then \(SA_{1}=\frac{18900}{36}=525\).

Answer:

\(525\space cm^{2}\)