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# The height of a cone is 15 cm. If its volume is $500\pi {\text{ c}}{{\text{m}}^3}$, then find the radius of its base.

Last updated date: 11th Aug 2024
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Hint: The formula to calculate the volume of a cone is, ${\text{Volume of Cone = }}\dfrac{1}{3}\pi {r^2}h$, where $r$ is the radius of the base of a cone and $h$ is the height of a cone. Apply this formula, and then use the given conditions to find the required value.

Given that the height $h$ of a cone is 15 cm and the volume of a cone is $500\pi {\text{ c}}{{\text{m}}^3}$.

Let us assume that the radius of the base of a cone is $r$.

We will use the formula of the volume of a cone is, ${\text{Volume of Cone = }}\dfrac{1}{3}\pi {r^2}h$, where $r$ is the radius of the base of a cone and $h$ is the height of a cone.

Substituting the values of $r$ and $h$ in the above formula of volume of a cone, we get

${\text{Volume of a cone = }}\dfrac{1}{3}\pi \times {r^2} \times 15 \\ = 5\pi {r^2} \\$

Since we know that the volume of the given cone is $500\pi {\text{ c}}{{\text{m}}^3}$.

Substituting the value of volume of a cone in the above equation, we get

$500\pi = 5\pi {r^2}$

Dividing the above equation by $5\pi$on each of the sides, we get

$\Rightarrow \dfrac{{500\pi }}{{5\pi }} = \dfrac{{5\pi {r^2}}}{{5\pi }} \\ \Rightarrow 100 = {r^2} \\ \Rightarrow {r^2} = 100 \\$

Taking square root on each of the sides in the above equation, we get

$\Rightarrow {r^2} = 100 \\ \Rightarrow r = \pm 10{\text{ }} \\$

Since the radius of a cone can never be negative, the negative value of $r$ is discarded.

Therefore, the radius of a cone is 10 cm.

Note: In solving these types of questions, you should be familiar with the formula of the volume of a cone. To make the calculations easier, the value of $\pi$ has not been substituted. Since $\pi$ occurs in the expression for the volume and its formula, hence, cancels out when they are equated and will help in reducing the calculations to a great extent.