Question

The volumes of two similar solids are 210 m3 and 1,680 m3. The surface area of the larger solid is 856 m2. What is the surface area of the smaller solid?

Answer 1

we know that

$scale \ factor^{3}=\frac {volume\ larger\ solid }{volume\ smaller\ solid}$

so

Find the value of the scale factor

$volume\ larger\ solid= 1,680\ m^{3} \\ volume\ smaller\ solid= 210\ m^{3}$

substitute the values in the formula

$scale \ factor^{3}=\frac {1,680 }{210}$

$scale \ factor^{3}=8$

$scale \ factor=\sqrt[3]{8} \\ scale \ factor= 2$

Find the surface area of the smaller solid

we know that

$scale \ factor^{2}=\frac {surface\ area\ larger\ solid }{surface\ area\ smaller\ solid}$

$surface\ area\ larger\ solid =856\ m^{2} \\ scale\ factor =2$

$surface\ area\ smaller\ solid= \frac{surface\ area\ larger\ solid}{scale \ factor^{2}}$

substitute the values

$surface\ area\ smaller\ solid= \frac{856}{2^{2}}$

$surface\ area\ smaller\ solid=214\ m^{2} }$

therefore

the answer is

The surface area of the smaller solid is equal to $214\ m^{2}$

Answer 2

$\bf \textit{ratio relations}\\ \begin{array}{llll} sides&\cfrac{s}{s}\\\\ Area&\cfrac{s^2}{s^2}\\\\ Volume&\cfrac{s^3}{s^3} \end{array}\qquad thus\\\\ -----------------------------\\\\ \cfrac{\textit{smaller volume}}{\textit{bigger volume}}\implies \cfrac{210}{1680}=\cfrac{s^3}{s^3}\implies \cfrac{210}{1680}=\left( \cfrac{s}{s} \right)^3 \\\\\\ \sqrt[3]{\cfrac{210}{1680}}=\cfrac{s}{s}\implies \cfrac{\sqrt[3]{210}}{\sqrt[3]{1680}}=\cfrac{s}{s}\\\\ -----------------------------$

$\bf now \\\\ \cfrac{\textit{smaller area}}{\textit{bigger area}}\implies \cfrac{A}{856}=\cfrac{s^2}{s^2}\implies \cfrac{A}{856}=\left( \cfrac{s}{s}\right)^2 \\\\\\\\ \textit{however, recall that }\sqrt[3]{\cfrac{210}{1680}}=\cfrac{s}{s}\qquad thus \\\\\\\\ \cfrac{A}{856}=\left( \sqrt[3]{\cfrac{210}{1680}}\right)^2\implies \cfrac{A}{856}=\cfrac{\sqrt[3]{210^2}}{\sqrt[3]{1680^2}}$

solve for "A", to get the surface area of the smaller solid

Answer 3

we know that

$scale \ factor^{3}=\frac {volume\ larger\ solid }{volume\ smaller\ solid}$

so

Find the value of the scale factor

$volume\ larger\ solid= 1,680\ m^{3} \\ volume\ smaller\ solid= 210\ m^{3}$

substitute the values in the formula

$scale \ factor^{3}=\frac {1,680 }{210}$

$scale \ factor^{3}=8$

$scale \ factor=\sqrt[3]{8} \\ scale \ factor= 2$

Find the surface area of the smaller solid

we know that

$scale \ factor^{2}=\frac {surface\ area\ larger\ solid }{surface\ area\ smaller\ solid}$

$surface\ area\ larger\ solid =856\ m^{2} \\ scale\ factor =2$

$surface\ area\ smaller\ solid= \frac{surface\ area\ larger\ solid}{scale \ factor^{2}}$

substitute the values

$surface\ area\ smaller\ solid= \frac{856}{2^{2}}$

$surface\ area\ smaller\ solid=214\ m^{2} }$

therefore

the answer is

The surface area of the smaller solid is equal to $214\ m^{2}$