A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is #42 # and the height of the cylinder is #1 #. If the volume of the solid is #225 pi#, what is the area of the base of the cylinder?

Question

A solid consists of a cone on top of a cylinder with a radius equal to that of the cone.  The height of the cone is #42  # and the height of the cylinder is #1   #.  If the volume of the solid is #225 pi#, what is the area of the base of the cylinder?

Aurora Ayala · Accepted Answer

This question is much easier than it sounds. It involves some basic algebra and simple calculations. Not all what you might expect!

Area of base = #15pi# What we already know is that the height of the cylinder and cone, as well as the total volume of the shape, are known. The cylinder and cone have the same radius, but the length is unknown. Write the following word equation down: vol of shape + vol of cylinder + vol cone Now use the formulae: #pir^2h + (pir^2H)/3 = 225pi# Factorise (common factor): #pir^2(h + H/3) = 225pi# Divide both sides by #pi# and substitute: #r^2(1 + 42/3) = 225# Simplify: #r^2 xx 15 = 225# This gives us: #r^2# = #225/15# = 15 We could calculate the radius, but it is not necessary, we need #r^2# to find the area of the base which is a circle. Area of the base of thecylinder = #pir^2 = 15pi# Let's check: #pir^2h + (pir^2H)/3# = #15pi(1) + (15pi(42))/3# =#15pi + 210pi# =#225pi#

Caroline Aucoin · Answer

undefined First, let's denote the radius of both the cone and the cylinder as $r$. The volume $V$ of the solid is given by:

$$ V = V_{\text{cone}} + V_{\text{cylinder}} $$

The volume of a cone $V_{\text{cone}}$ with radius $r$ and height $h$ is given by:

$$ V_{\text{cone}} = \frac{1}{3} \pi r^2 h $$

The volume of a cylinder $V_{\text{cylinder}}$ with radius $r$ and height $h$ is given by:

$$ V_{\text{cylinder}} = \pi r^2 h $$

Given that the height of the cone ($h_{\text{cone}}$) is 42 and the height of the cylinder ($h_{\text{cylinder}}$) is 1, we have:

$$ V = \frac{1}{3} \pi r^2 (42) + \pi r^2 (1) = 225\pi $$

$$ \frac{1}{3} \pi r^2 (42) + \pi r^2 = 225\pi $$

$$ \frac{42}{3} r^2 + r^2 = 225 $$

$$ 14r^2 + r^2 = 225 $$

$$ 15r^2 = 225 $$

$$ r^2 = \frac{225}{15} = 15 $$

$$ r = \sqrt{15} $$

The area of the base of the cylinder is given by $A_{\text{cylinder}} = \pi r^2$:

$$ A_{\text{cylinder}} = \pi (\sqrt{15})^2 = \pi \times 15 = 15\pi $$

So, the area of the base of the cylinder is $15\pi$.