The volume of two spheres are in the ratio

Asked by admin @ 23/08/2021 in Math viewed by 325 People

The volume of two spheres are in the ratio 64:27. Find the ratio of their surface areas.

Answered by admin @ 23/08/2021

Answer:

The ratio of the surface areas is 16 : 9.

Step-by-step explanation:

To find the ratio of the surface areas, first we have to find the surface areas with their volumes.

Radius of big sphere = R

Radius of small sphere = r.

Volume of bigger sphere $\bold{=\frac{4}{3} \pi R^{3}}$

Volume of smaller sphere $\bold{=\frac{4}{3} \pi r^{3}}$

Given,

Volume of bigger sphere : Volume of smaller sphere = 64 : 27.

$\begin{array}{l}{\frac{\frac{4}{3} \pi R^{3}}{\frac{4}{3} \pi r^{3}}=\frac{64}{27}} \\ {\Rightarrow>\frac{R^{3}}{r^{3}}=\frac{64}{27}}\end{array}$

$\begin{aligned}=& \sqrt[3]{\frac{R^{3}}{r^{3}}}=\sqrt[3]{\frac{64}{27}} \\ &=>\frac{R}{r}=\frac{4}{3} \end{aligned}$

Surface area of bigger sphere $=4 \pi R^{2}$

Surface area of smaller sphere $=4 \pi r^{2}$

Hence, Surface area of bigger sphere: Surface area of smaller sphere = $\begin{array}{l}\bold{{=4 \pi R^{2} : 4 \pi r^{2}}} \\ {=\frac{4 \pi R^{2}}{4 \pi r^{2}}=\frac{R^{2}}{r^{2}}} \\ {=\left(\frac{4}{3}\right)^{2}=\frac{16}{9}}\end{array}$

Thus, the ratio of their surface areas = 16 : 9

The volume of two spheres are in the ratio

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