Question

Using clay, a student made a right circular cone of height $48$cm and base radius $12$cm. Another student reshapes it in the form of a sphere. Find the radius of the sphere.
A) $12\sqrt[3]{3}$
B) $3\sqrt[3]{3}$
C) $2\sqrt[3]{{36}}$
D) $12$

Answer 1

Hint: The first thing to note here is we have been given two different three-dimensional shapes formed with the clay by two different people. The clay used is in the same amount that means we have the same volume of clay available. We will use the formulae for volume of cone and sphere to compare the quantities and then determine the radius.

Complete step-by-step answer:
It is given that the height of the cone is $48$ cm and the radius of the base is $12$ .
The same amount of clay is used to form the sphere without making any change in the amount of clay.
This implies that the volume of the cone formed and the volume of the cylinder formed are one and the same.
For a cylinder with height $h$ and the base radius $r$ , the volume of the cylinder is given by the following expression.
${V_1} = \pi {r^2}h$
In this case it is given that the height of the cone is $48$ cm and the radius of the base is $12$ .
Therefore, the volume of the cylinder is:
${V_1} = \pi {\left( {12} \right)^2}48$
Now for a sphere with radius $R$, the volume of the sphere is given by:
${V_2} = \dfrac{4}{3}\pi {R^3}$
We have already established for the given case that ${V_1} = {V_2}$ .
Therefore,
$\dfrac{4}{3}\pi {R^3} = \pi {\left( {12} \right)^2}48$
Rearranging the given terms, we write:
${R^3} = \dfrac{3}{4} \times {\left( {12} \right)^2} \times 48$
Taking cube root, we get,
\[R = 12\sqrt[3]{3}\]
Therefore, the radius of the sphere is $12\sqrt[3]{3}$ .

Hence, the correct option is A.

Note: We have to form two different equations so that we can have only one unknown which can be determined by simplifying the formed equations. We also need to understand the fact that the clay is the same so the volume will be the same. We need to use the formula for volume of both the quantities.