Question

Find the ratio of the volume of sphere $A$ to sphere $B$, if the ratio of the surface area of sphere $A$ to the surface area of sphere $B$ is $729:1$.

• A. $27:1$
• B. $81:1$
• C. $19,683:1$
• D. $26,224:1$
• E. $531,441:1$

Answer 1

Answer: (C)

$19,683:1$

Let the radius of sphere $A$ be $a$ and the radius of sphere $B$ be $b$

Given, $\frac{4\pi a^2}{4\pi b^2}=729$

$\Rightarrow \frac{a^2}{b^2}={\left(\frac{a}{b}\right)}^2=729={27}^2$

$\Rightarrow \frac{a}{b}=27$

Ratio of volume of $A$ to volume of $B$ is $\frac{\frac{4}{3}\pi a^3}{\frac{4}{3}\pi b^3}$

$={\left(\frac{a}{b}\right)}^3={27}^3$

$=19683$