Question

drag each label to the correct location on the table. not all labels will be used.
the perimeter of base dhgc is 30 inches, and the perimeter of base swr is 20 inches. complete the table with the ratios of the heights, surface areas, and volumes of these two similar rectangular prisms. assume the ratios are written in the form figure 1 : figure 2.
ratio of heights
ratio of surface areas
ratio of volumes

Explanation:

Step1: Find the ratio of heights

For similar solids, the ratio of corresponding linear measures (like perimeters of similar bases) is equal to the ratio of their heights.
Let the ratio of heights of Figure 1 to Figure 2 be \(k\).
We know that if the perimeter of base \(DHGC\) (\(P_1 = 30\)) and the perimeter of base \(SWR\) (\(P_2=20\)), then \(k=\frac{P_1}{P_2}\) (since for similar solids, the ratio of corresponding linear - measures is consistent).
\(k = \frac{30}{20}=\frac{3}{2}\)

Step2: Find the ratio of surface areas

For similar solids, if the ratio of heights (or any linear measure) is \(a:b\), the ratio of their surface areas is \(a^{2}:b^{2}\).
Since the ratio of heights \(k=\frac{3}{2}\), the ratio of surface areas \(k_{SA}=(\frac{3}{2})^{2}=\frac{9}{4}\)

Step3: Find the ratio of volumes

For similar solids, if the ratio of heights (or any linear measure) is \(a:b\), the ratio of their volumes is \(a^{3}:b^{3}\).
Since the ratio of heights \(k = \frac{3}{2}\), the ratio of volumes \(k_{V}=(\frac{3}{2})^{3}=\frac{27}{8}\)

Answer:

• Ratio of Heights: \(\frac{3}{2}\)
• Ratio of Surface Areas: \(\frac{9}{4}\)
• Ratio of Volumes: \(\frac{27}{8}\)