Question

these solids are similar. find the surface area of the larger solid.
r = 4 m
sa = 176 m²
r = 5 m
sa = ? m²

Explanation:

Step1: Find the ratio of the radii

The ratio of the radii of the two similar cones is \(\frac{r_{1}}{r_{2}}=\frac{4}{5}\) (where \(r_{1} = 4m\) is the radius of the smaller cone and \(r_{2}=5m\) is the radius of the larger cone). For similar solids, if the ratio of corresponding linear dimensions (like radius) is \(a:b\), the ratio of their surface - areas is \(a^{2}:b^{2}\). So the ratio of the surface - areas \(SA_{1}:SA_{2}\) is \((\frac{4}{5})^{2}=\frac{16}{25}\), where \(SA_{1}\) is the surface area of the smaller cone and \(SA_{2}\) is the surface area of the larger cone.

Step2: Set up a proportion to find \(SA_{2}\)

We know that \(\frac{SA_{1}}{SA_{2}}=\frac{16}{25}\), and \(SA_{1} = 176m^{2}\). Substituting the values into the proportion \(\frac{176}{SA_{2}}=\frac{16}{25}\). Cross - multiply: \(16\times SA_{2}=176\times25\). Then \(SA_{2}=\frac{176\times25}{16}\).
First, calculate \(176\div16 = 11\). Then \(11\times25=275\).

Answer:

\(275\)