Question

A cube and a sphere have equal volumes. Then, the ratio of their total surface area is

• A. $\sqrt[3]{36}:\sqrt[3]{3\pi }$
• B. $\sqrt[3]{34}:\sqrt[3]{3{\pi }^2}$
• C. $\sqrt[3]{318}:\sqrt[3]{3{\pi }^2}$
• D. $\sqrt[3]{39}:\sqrt[3]{3\pi }$

Answer 1

The correct option is A $\sqrt[3]{36}:\sqrt[3]{3\pi }$

Let r and a be the radius of the sphere and side of the cube respectively.
Volume of cube = Volume of sphere
$\therefore a^3=\frac{4}{3}\pi r^3$
$\Rightarrow \frac{a^3}{r^3}=\frac{4\pi }{3}$
$\Rightarrow \left(\frac{a}{r}\right)=\sqrt[3]{3\frac{4\pi }{3}}$ ....(i)
$\therefore \text{Ratio of their total surface area}=\text{TSA of cube : TSA of sphere}$
$=6a^2:4\pi r^2$
$=\frac{6a^2}{4\pi r^2}$
$=\frac{3}{2\pi }\times {\left(\frac{a}{r}\right)}^{\frac{2}{3}}$
$=\frac{3}{2\pi }\times {\left(\frac{4\pi }{3}\right)}^{\frac{2}{3}}$ [From (i)]
$=3^{1-\frac{2}{3}}\times (2{)}^{\frac{4}{3}-1}\times (\pi {)}^{\frac{2}{3}-1}$
$=3^{\frac{1}{3}}\times 2^{\frac{1}{3}}\times {\pi }^{\frac{-1}{3}}$
$={\left(\frac{3\times 2}{\pi }\right)}^{\frac{1}{3}}$
$=\sqrt[3]{3\frac{6}{\pi }}$
$=\sqrt[3]{36}:\sqrt[3]{3\pi }$
Hence, the correct answer is option (a).