Question

Volume of a hollow sphere is ${\displaystyle \frac{11352}{7}}{\text{ cm}}^3$ . If the outer radius is 8 cm, find the inner radius of the sphere. (Take $\pi ={\displaystyle \frac{22}{7}}$ )

Answer 1

Hint: A sphere is a ball-shaped three-dimensional object.
The volume of a sphere of radius r units is ${\displaystyle \frac{4}{3}}\pi r^3$ cubic units.
The volume of the material used in making a hollow sphere = Volume of outer sphere - Volume of inner sphere.
Assume the inner radius to be x cm, form an equation and solve.

Complete step by step answer:
Let's say that the inside radius of the hollow sphere is x cm and the outside radius is $y=8\text{ cm}$ .

Using the formula $V={\displaystyle \frac{4}{3}}\pi r^3$ , the volume of the outside sphere is ${\displaystyle \frac{4}{3}}\pi 8^3$ and the volume of the inside sphere is ${\displaystyle \frac{4}{3}}\pi x^3$ .
The volume of the hollow sphere (shaded part) will be ${\displaystyle \frac{4}{3}}\pi 8^3-{\displaystyle \frac{4}{3}}\pi x^3$ .
According to the question:
 ${\displaystyle \frac{4}{3}}\pi 8^3-{\displaystyle \frac{4}{3}}\pi x^3={\displaystyle \frac{11352}{7}}$
Taking out the common factors ${\displaystyle \frac{4}{3}}\pi$ and using $\pi ={\displaystyle \frac{22}{7}}$ , we get:
⇒ ${\displaystyle \frac{4}{3}}\times {\displaystyle \frac{22}{7}}\times (8^3-x^3)={\displaystyle \frac{11352}{7}}$
⇒ $8^3-x^3={\displaystyle \frac{11352}{7}}\times {\displaystyle \frac{7}{22}}\times {\displaystyle \frac{3}{4}}$
Note that 11352 is multiple of 11, because $(1+3+2)-(1+5)=6-6=0$ . Dividing 11352 by 22 and cancelling out the 7's, we get:
⇒ $512-x^3=516\times {\displaystyle \frac{3}{4}}$
⇒ $512-x^3=129\times 3$
⇒ $512-x^3=387$
⇒ $x^3=512-387$
⇒ $x^3=125$
Since, $5\times 5\times 5=125$ , we get:
⇒ $x=5$

∴ The inner radius of the sphere is 5 cm.

Note: The same idea can be applied to solids of other shapes. The surface area of a sphere is $4\pi r^2$ sq. units. The half of a sphere is also called a hemi-sphere.