Question

if the volumes of two spheres are
64:27, then find the ratio
of their surface areas​

Answer 1

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

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Answer 2

Answer:-

Ratio of volume of two spheres = 64 : 27

4/3 πR³/4/3 πr³

R³/r³ = 64/27

R = ³√64 = 4 units

r = ³√27 = 3 units

Now we have to find the ratio of their Surface Area

4 πR²/4 πr²

R²/r²

= (4)²/(3)²

= 16/9

Hence, the ratio is 16 : 9