Question

If the surface areas of two similar figured are 25cm^2 and 36cm^2, and the volume of the larger figure is 216cm^3, what is the volume of the smaller figure?

Answer 1

If two figures are similar, then there is a relation between their areas and volumes:

`\begin{gathered} (A_1)/(A_2)=k^2...(1) \\ \\ (V_1)/(V_2)=k^3...(2) \end{gathered}`

Where k is a constant. From the problem, we identify:

`\begin{gathered} A_1=25\text{ cm}^2 \\ A_2=36\text{ cm}^2 \\ V_2=216\text{ cm}^3 \end{gathered}`

Using equation (1):

`\begin{gathered} (25)/(36)=k^2 \\ \\ \Rightarrow k=(5)/(6) \end{gathered}`

Finally, using (2) to find the volume of the smaller figure:

`\begin{gathered} (V_1)/(216)=(5^3)/(6^3) \\ \\ (V_1)/(216)=(125)/(216) \\ \\ \therefore V_1=125\text{ cm}^2 \end{gathered}`