Question

Sphere 1 has surface area A₁ and volume V₁, sphere 2 has surface area A₂ and volume V₂. If the radius of sphere 2 is six times the radius of sphere 1, what is the ratio $\frac{A2}{A1}$ of the areas? What is the ratio $\frac{V2}{V1}$ of the volumes?

Answer 1

Let , radius of sphere 1 is r .

So , radius of sphere 2 is 6r .

Surface area of sphere is given by :

$A=4\pi r^2$

So ,

$\dfrac{A_2}{A_1}=\dfrac{4\pi(6r)^2}{4\pi r^2}\\\\\dfrac{A_2}{A_1}=36$

Volume is given by :

$V=\dfrac{4}{3}\pi r^3$

Ratio of sphere 2 by sphere 1 is given by :

$\dfrac{V_2}{V_1}=\dfrac{\dfrac{4}{3}\pi (6r)^3}{\dfrac{4}{3}\pi r^3}\\\\\dfrac{V_2}{V_1}=216$

Therefore ,  the ratio of area and volume is 36 and 216 respectively .

Hence , this is the required solution .