Question

the volumes of two similar solids are 210 m³ and 1,680 m³. the surface area of the larger solid is 856 m². what is the surface area of the smaller solid? 107 m² 214 m² 1084 m² 6948 m²

Explanation:

Step1: Find the ratio of volumes

The ratio of the volumes of two similar solids is \(\frac{V_{small}}{V_{large}}=\frac{210}{1680}=\frac{1}{8}\)

Step2: Find the ratio of side lengths

Since for similar solids, if the ratio of volumes is \(\frac{V_1}{V_2}=k^{3}\) (where \(k\) is the ratio of side lengths), then \(k = \sqrt[3]{\frac{1}{8}}=\frac{1}{2}\)

Step3: Find the ratio of surface areas

Since for similar solids, if the ratio of side lengths is \(k\), the ratio of surface areas is \(k^{2}\). Let the surface area of the smaller solid be \(S\). We know \(\frac{S}{S_{large}}=k^{2}\), and \(k = \frac{1}{2}\), so \(k^{2}=\frac{1}{4}\)

Step4: Solve for the surface area of the smaller solid

We have \(\frac{S}{856}=\frac{1}{4}\), then \(S=\frac{856}{4}=214\)

Answer:

B. \(214m^{2}\)