Question

the two triangular prisms shown are similar. the dimensions of the larger prism were multiplied by a scale factor of \\(\frac{4}{5}\\) to create the smaller prism. when the large prism was reduced, the surface area changed by a factor of \\(\frac{64}{125}\\), \\(\frac{16}{25}\\), \\(\frac{4}{5}\\), \\(\frac{10}{8}\\).

Explanation:

Step1: Recall the formula for the ratio of surface areas of similar solids

If the scale factor of two similar solids is \(k\), the ratio of their surface areas is \(k^{2}\).

Step2: Identify the scale factor

The scale factor \(k=\frac{4}{5}\) (since the larger prism was reduced by a factor of \(\frac{4}{5}\) to get the smaller prism).

Step3: Calculate the ratio of the surface areas

Using the formula \(k^{2}\), substitute \(k = \frac{4}{5}\). Then \(k^{2}=(\frac{4}{5})^{2}=\frac{4\times4}{5\times5}=\frac{16}{25}\)

Answer:

\(\frac{16}{25}\) (the second option)