Question

. The volumes of the two spheres are in the ratio 64: 27. Find the ratio of their surface area.

Answer 1

Let radius of first sphere be 'a'.

Let radius of second sphere be 'b'.

Volume of first sphere is given by

$v(a) = \frac{4}{3} \pi {a}^{3}$

Volume of second sphere is given by

$v(b) = \frac{4}{3} \pi {b}^{3}$

$\frac{v(a)}{v(b)} = \frac{ \frac{4}{3}\pi {a}^{3} }{ \frac{4}{3}\pi {b}^{3} } = { ( \frac{a}{b} )}^{3} = \frac{64}{27}$

$\frac{a}{b} = \frac{4}{3}$

Surface area of first sphere is given by

$s(a) = 4\pi {a}^{2}$

Surface area of second sphere is given by

$s(b) = 4\pi {b}^{2}$

$\frac{s(a)}{s(b)} = \frac{4\pi {a}^{2} }{4\pi {b}^{2} } = {( \frac{a}{b} ) }^{2} = {( \frac{4}{3} )}^{2} = \frac{16}{9}$

Therefore,

Ratio of surface area is 16:9

Answer 2

The Ratio of Surface area is 16 : 9.

Step-by-step explanation:

Given:

The volumes of the two spheres are in the ratio 64: 27

$\dfrac{V_{1}}{V_{2}}=\dfrac{64}{27}$

let r1 be the radius of one Sphere and

r2 be the radius of second sphere

To find:

Ratio of their surface area = ?

$\dfrac{S.A_{1}}{S.A_{2}}=?$

Solution:

Volume of sphere is given by formula

$\textrm{Volume of sphere}=\dfrac{4}{3}\pi(radius)^{3}$

Therefore the Ratio of Volume will be

$\dfrac{V_{1}}{V_{2}}= \dfrac{\dfrac{4}{3}\pi(r1)^{3} }{\dfrac{4}{3}\pi(r2)^{3}}$

substituting the values we get

$\dfrac{64}{27}= \dfrac{(r1)^{3}}{(r2)^{3}} \\Cube\ Rooting\ we\ get\\\dfrac{r1}{r2}=\dfrac{4}{3}$

Now, Surface area of sphere is given by

$\textrm{Surface area of sphere}=4\pi (radius)^{2}$

So the surface area Ratio will be

$\dfrac{S.A_{1}}{S.A_{2}}=\dfrac{4\pi(r1)^{2} }{4\pi(r2)^{2} } \\\\\dfrac{S.A_{1}}{S.A_{2}}=\dfrac{(r1)^{2} }{(r2)^{2} } \\Substituting\\\dfrac{S.A_{1}}{S.A_{2}}=\dfrac{4^{2} }{3^{2} }=\dfrac{16}{9} \\\dfrac{S.A_{1}}{S.A_{2}}=\dfrac{16}{9}$

The ratio of their surface area is 16 : 9