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 #18 # and the height of the cylinder is #1 #. If the volume of the solid is #84 pi#, what is the area of the base of the cylinder?

Answer 1

#12pi#

Assume the radius of the cylinder/cone as r, height of cone as #h_1#, height of cylinder as #h_2#
Volume of the cone part of solid = #(pi*r^2*h_1)/3#
Volume of the cylinder part of solid = # pi*r^2 * h_2#

What we currently possess is:

#h_1# = 18,#h_2# = 1
#(pi*r^2*h_1)/3# + # pi*r^2 * h_2# = #84*pi#
#(pi*r^2*18)/3# + # pi*r^2 * 1# = #84*pi#
# pi*r^2 * 6# + # pi*r^2 * 1# = #84*pi#
# pi*r^2 * 7# = #84*pi#
# r^2 # = #84/7# = 12
Area of the base of the cylinder = #pi*r^2# = #pi*12# = #12pi#
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Answer 2

To find the area of the base of the cylinder, we first need to find the radius of both the cone and the cylinder. Since the radius of the cylinder is equal to the radius of the cone, we'll denote it as ( r ).

Given:

  • Height of the cone (( h_{\text{cone}} )) = 18 units
  • Height of the cylinder (( h_{\text{cylinder}} )) = 1 unit
  • Volume of the solid = 84π cubic units

The volume ( V ) of the solid consisting of a cone on top of a cylinder is given by the sum of the volumes of the cone and the cylinder.

The volume ( V ) of a cone is given by: [ V_{\text{cone}} = \frac{1}{3} \pi r^2 h ]

The volume ( V ) of a cylinder is given by: [ V_{\text{cylinder}} = \pi r^2 h ]

Given ( V_{\text{solid}} = 84\pi ), we can write: [ V_{\text{solid}} = V_{\text{cone}} + V_{\text{cylinder}} ]

Substituting the formulas for ( V_{\text{cone}} ) and ( V_{\text{cylinder}} ), we get: [ 84\pi = \frac{1}{3} \pi r^2 \times 18 + \pi r^2 \times 1 ]

From this equation, we can solve for ( r ). Once we find ( r ), we can use it to find the area of the base of the cylinder using the formula for the area of a circle: [ A_{\text{cylinder base}} = \pi r^2 ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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