It is given that two similar solids have surface areas of 48 m² and 147 m², and the smaller solid has a volume of 34 m³.
It is required to find the volume of the larger solid.
Recall that the if the scale factor of similar solids is a/b, then the ratio of their areas is the square of the scale factor:
[tex]\frac{\text{ Area of smaller solid}}{\text{ Area of larger solid}}=\frac{a^2}{b^2}[/tex]Substitute the given areas into the equation:
[tex]\frac{48}{147}=\frac{a^2}{b^2}[/tex]Find the scale factor a/b:
[tex]\begin{gathered} \text{ Swap the sides of the equation:} \\ \Rightarrow\frac{a^2}{b^2}=\frac{48}{147} \\ \text{ Reduce the fraction on the right with }3: \\ \Rightarrow\frac{a^2}{b^2}=\frac{16}{49} \\ \text{ Take the square root of both sides:} \\ \Rightarrow\frac{a}{b}=\frac{4}{7} \end{gathered}[/tex]Recall that if the scale factor of two similar solids is a/b, then the ratio of their volumes is the cube of the scale factor:
[tex]\frac{\text{ Volume of smaller solid}}{\text{ Volume of larger solid}}=\left(\frac{a}{b}\right)^3[/tex]Let the volume of the larger solid be V and substitute the given value for the volume of the smaller solid:
[tex]\frac{34}{V}=\left(\frac{a}{b}\right)^3[/tex]Substitute a/b=4/7 into the proportion:
[tex]\begin{gathered} \frac{34}{V}=\left(\frac{4}{7}\right)^3 \\ \\ \Rightarrow\frac{34}{V}=\frac{4^3}{7^3} \\ \\ \Rightarrow\frac{34}{V}=\frac{64}{343} \end{gathered}[/tex]Find the value of V in the resulting proportion:
[tex]\begin{gathered} \text{ Cross multiply:} \\ 64V=343\cdot34 \\ \text{ Divide both sides by }64: \\ \Rightarrow\frac{64V}{64}=\frac{343\cdot34}{64} \\ \Rightarrow V\approx182.22\text{ m}^3 \end{gathered}[/tex]Answers:
The required proportion is 34/V =64/343.
The volume of the larger solid is about 182.22 m³.
