Question Video: Approximating the Height of a Cone from Its Volume Mathematics • Second Year of Preparatory School

Video Transcript

The volume of a right circular cone is given by 𝑉 is equal to one-third 𝜋𝑟 squared ℎ, where 𝜋 is approximately equal to 22 over seven. If the volume of a right circular cone equals 462 cubic centimeters and the radius 𝑟 of its base is seven centimeters, find the height of the cone.

In this question, we are given a formula for the volume of a right circular cone. We are told that the volume of a right circular cone is 462 cubic centimeters, that is, the value of 𝑉. We are told that the radius of the base is seven centimeters, that is, the value of 𝑟. And we want to find the height ℎ of the cone.

Let’s start by sketching the information. First, we can start with a right circular cone. That is, the base of the cone is a circle and the vertex lies above the center of the circle. The height is the perpendicular distance between the vertex and the center of the base. And the radius is the distance between the center of the circle and its circumference. To find the value of ℎ, we need to use the given values and the given formula.

We can start by substituting 𝑉 is equal to 462, 𝜋 is approximately equal to 22 over seven, and 𝑟 is equal to seven into the volume formula. This gives us that 462 is approximately equal to one-third times 22 over seven times seven squared times ℎ. In the order of operations, we evaluate exponents before multiplication and division. So we start by evaluating seven squared. This is equal to seven times seven, which is 49. This gives us the following equation.

We can now simplify the right-hand side of the equation by multiplying the coefficients of ℎ. We get a coefficient of one times 22 times 49 over three times seven. We can simplify further by canceling the shared factor of seven in the numerator and denominator. We can then calculate that 22 times seven is 154. So we have that 462 is equal to 154 over three multiplied by ℎ.

We want to find the value of ℎ. So we need to isolate ℎ on the right-hand side. We can do this by dividing both sides through by 154 over three. We can then recall that dividing by a fraction is the same as multiplying by its reciprocal. So we can instead multiply by three over 154. This gives us that ℎ is approximately equal to 462 times three over 154.

We can then note that there is a shared factor of 154 in the numerator and denominator, since three times 154 is equal to 462. This means that the height of the cone is three times three, which is equal to nine. We can also give this the units of centimeters, since all of the measurements use centimeters. Hence, the height of the cone is approximately nine centimeters.