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 #39 # and the height of the cylinder is #17 #. If the volume of the solid is #90 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 #39  # and the height of the cylinder is #17   #.  If the volume of the solid is #90 pi#, what is the area of the base of the cylinder?

Aurora Ayala · Accepted Answer

This question involves some basic algebra and really simple calculations. It has a very good feel to it as you work through it.

Area of the base = #3pi# 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 radius of the cylinder and the cone is the same, but we do not know the length of the radius. Write the following word equation down: vol of shape + vol of cylinder + vol cone Now use the formulae: #pir^2h + (pir^2H)/3 = 90pi# Factorise (common factor): #pir^2(h + H/3) = 90pi# Divide both sides by #pi# and substitute: #r^2(17 + 39/3) = 90# Simplify: #r^2 xx 30 = 90# This gives us: #r^2# = #90/30# = 3 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 = #pir^2 = 3pi# Let's check: #pir^2h + (pir^2H)/3# = #3pi(17) + (3pi(39))/3# =#51pi + 39pi# =#90pi#

Ella Armstrong · Answer

undefined To find the area of the base of the cylinder, we can first calculate the combined volume of the cone and the cylinder using the given information about their dimensions and the total volume of the solid. Then, we can subtract the volume of the cone from the total volume to find the volume of the cylinder. Finally, using the formula for the volume of a cylinder, we can solve for the radius of the cylinder and then calculate the area of its base.

1. Calculate the volume of the cone:
   The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$, where $r$ is the radius and $h$ is the height.
   Substituting the given values: $V_{\text{cone}} = \frac{1}{3} \pi (r^2)(39)$.

2. Calculate the volume of the cylinder:
   The formula for the volume of a cylinder is $V = \pi r^2 h$, where $r$ is the radius and $h$ is the height.
   Substituting the given values: $V_{\text{cylinder}} = \pi (r^2)(17)$.

3. Since the total volume of the solid is given as $90 \pi$, we have:
   $90 \pi = V_{\text{cone}} + V_{\text{cylinder}}$.

4. Substitute the expressions for $V_{\text{cone}}$ and $V_{\text{cylinder}}$ into the equation and solve for $r$.

5. Once you find the radius $r$, use it to calculate the area of the base of the cylinder using the formula $A = \pi r^2$.

Following these steps, you can find the area of the base of the cylinder.