### Video Transcript

A cone has a perpendicular height
of 92 inches and a volume of 420𝜋 cubic inches. Work out the radius of the cone,
giving your answer to the nearest inch.

In order to answer this question,
we need to recall the formula for the volume of a cone. This is equal to one-third 𝜋𝑟
squared multiplied by ℎ, where 𝑟 is the radius of the circular base and ℎ is the
perpendicular distance from the apex of the cone to the base. In this question, we are given both
the height and the volume and we are asked to calculate the radius. To do this, we will begin by
rearranging the formula 𝑉 is equal to one-third 𝜋𝑟 squared ℎ to make 𝑟 the
subject.

We begin by multiplying both sides
of our equation by three such that three 𝑉 is equal to 𝜋𝑟 squared ℎ. Next, we divide both sides by 𝜋ℎ
such that 𝑟 squared is equal to three 𝑉 divided by 𝜋ℎ. Finally, we take the square root of
both sides of the equation. As the radius must be positive, 𝑟
is equal to the square root of three 𝑉 divided by 𝜋ℎ. We can now substitute 420𝜋 for 𝑉
and 92 for ℎ. Three multiplied by 420𝜋 is
1260𝜋, and 𝜋 multiplied by 92 is 92𝜋. Underneath the square root, we can
divide the numerator and denominator by 𝜋. As 1260 and 92 are both divisible
by four, 𝑟 is equal to the square root of 315 over 23. Typing this into the calculator
gives us 3.7007 and so on.

As we are asked to give our answer
to the nearest inch, we need to consider the digit in the tenths column. We can therefore conclude that a
cone with a perpendicular height of 92 inches and a volume of 420𝜋 cubic inches
will have a radius of four inches correct to the nearest inch. We could check this answer by
substituting the value of 𝑟 back in to the original formula.