Question

the areas of the bases of similar rectangular prisms are 25 square inches and 64 square inches. what is the ratio of the volumes of the rectangular prisms? 5 to 8 15 to 24 75 to 192 125 to 512

Explanation:

Step1: Find the ratio of side lengths

For similar solids, if the ratio of the areas of corresponding faces is \(A_1:A_2\), and the ratio of side lengths is \(s_1:s_2\), then \(A_1:A_2=(s_1:s_2)^2\).
Given \(A_1 = 25\) and \(A_2=64\), so \((s_1:s_2)^2=\frac{25}{64}\), then \(s_1:s_2=\sqrt{\frac{25}{64}}=\frac{5}{8}\).

Step2: Find the ratio of volumes

For similar solids, if the ratio of side lengths is \(s_1:s_2\), the ratio of volumes \(V_1:V_2=(s_1:s_2)^3\).
Since \(s_1:s_2 = \frac{5}{8}\), then \(V_1:V_2=(\frac{5}{8})^3=\frac{125}{512}\).

Answer:

125 to 512