Question

The surface areas of the two spheres are in the ratio 1:4.
Find the ratio of their volume?

Answer 1

Hint: We use the formula of surface area of sphere is \[4 \pi {r^2}\] and volume of sphere is \[\dfrac{4}{3}\pi {r^3}\]. Now, we take the radius of one sphere as r and the other as r’, now we take the ratio of there surface area and equate it with the given ratio. We get the ratio of the radius of the two spheres, now to calculate the ratio of volumes of the two spheres we use the ratio of the radius of the two spheres, and on substituting its value we get the required answer.

Complete step by step solution: Ratio of surface area of the two spheres are given as per using the condition given in the question
\[\dfrac{{\left( {4\pi {r^2}} \right)}}{{\left( {4\pi {{\left( {r'} \right)}^2}} \right){\text{ }}}} = \dfrac{1}{4}\] (1)
On simplifying (1) ,
$\dfrac{{{r^2}}}{{{{\left( {r'} \right)}^2}}} = {\left( {\dfrac{1}{2}} \right)^2}$
Hence the radius ratio is obtained and use that for calculation of ratio of volume of sphere
$\dfrac{r}{{r'}} = \dfrac{1}{2}$
Now the ratios of the volume of the spheres can be given as
\[\dfrac{{\dfrac{4}{3}\pi {r^3}}}{{\dfrac{4}{3}\pi {{\left( {r'} \right)}^3}}} = \dfrac{{{r^3}}}{{{{\left( {r'} \right)}^3}}}\] …(3)
Using (2) in (3) means using the radius ratio obtained in the above equation in order to calculate the volume ratio of spheres will be cubes time of the radius ratio.
\[\dfrac{{{r^3}}}{{{{\left( {r'} \right)}^3}}} = {\left( {\dfrac{1}{2}} \right)^3} = \dfrac{1}{8}\]

Hence, ratio of volume of spheres is \[\dfrac{1}{8}\].

Note: Compare the surface area to the volume ratio in the above case as solved in the solution. When an object/cell is very small, it has a large surface area to volume ratio, while a large object/ cell has a small surface area to volume ratio. When a cell grows, its volume increases at a greater rate than its surface area, therefore it's SA: V ratio decreases.