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 #60 # and the height of the cylinder is #15 #. If the volume of the solid is #7 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 #60  # and the height of the cylinder is #15   #.  If the volume of the solid is #7  pi#, what is the area of the base of the cylinder?

Caroline Aucoin · Accepted Answer

The area of the base of the cylinder is #(7pi)/100#.

The volume of the solid is found by adding the volume of the cylinder to the volume of the cone. The volume of a cylinder is given by #V = pir^2h#. The volume of the cone is given by #V = (pir^2h)/3#. So the volume of the entire solid is given by: #V = pir^2h + (pir^2h)/3# By using the Distributive Property in reverse, this can be changed to become: #V = pir^2h(1 + 1/3)# which consequently turns into: #V = (4pir^2h)/3# Since we are looking for the area of the base, which is a circle, we need to isolate the part of the formula which gives that area. The area of a circle is given by #A = pir^2#, so we need to isolate that part of the simplified volume formula. #3*V = 3*(4pir^2h)/3# #3V = 4pir^2h# #(3V)/(4h) = (4pir^2h)/(4h)# #(3V)/(4h) = pir^2# Consequently, the following formula yields the area of the cylinder's base: #A = (3V)/(4h)# Substitute the values you know for #V# and #h# and simplify as possible. {#h = 60 + 15 = 75#} #A = (3*7pi)/(4*75)# #A = (21pi)/300# #A = (7pi)/100# Since the volume in the problem is left in terms of #pi#, it is acceptable to give the area of the base in terms of #pi# as well.

Theodore Abad · Answer

undefined Given that the volume of the solid is 7π and the solid consists of a cone on top of a cylinder, we can set up an equation for the volume of the solid.

The volume $ V $ of the solid is the sum of the volumes of the cone and the cylinder:

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

The volume $ V_{\text{cone}} $ of the cone is given by:

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

where $ r $ is the radius of the cone and $ h $ is the height of the cone.

The volume $ V_{\text{cylinder}} $ of the cylinder is given by:

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

where $ r $ is the radius of the cylinder and $ h $ is the height of the cylinder.

Given that the height of the cone is 60 and the height of the cylinder is 15, and the radius of the cone is equal to that of the cylinder, we can express the volumes in terms of the radius $ r $:

$$ V_{\text{cone}} = \frac{1}{3}\pi r^2 \times 60 $$
$$ V_{\text{cylinder}} = \pi r^2 \times 15 $$

Substituting these expressions into the equation for the volume of the solid and setting it equal to 7π:

$$ 7\pi = \frac{1}{3}\pi r^2 \times 60 + \pi r^2 \times 15 $$

Solve for $ r $ using this equation, then use the value of $ r $ to calculate the area of the base of the cylinder, which is $ \pi r^2 $.