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 #12 # and the height of the cylinder is #28 #. If the volume of the solid is #72 pi#, what is the area of the base of the cylinder?

Answer 1

Area of base #= pir^2 = (9pi)/4#

#color(red)("Total volume = volume of cone + volume of cylinder" = 72 pi)#

volume of cone #= 1/3 pi r^2 h = 12/3pir^2 = 4*pir^2#

Volume of cylinder# = pi r^2 h = pi r^2 * 28#

Total volume# = 32pir^2 = 72 pi#

#cancelpir^2 = 72/32cancel pi=9/4#

#=>r ^2 = 9/4#

Area of base #= pir^2 = (9pi)/4#

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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 that of the cone, we can denote it as ( r ).

The volume of the solid is given by the sum of the volumes of the cone and the cylinder:

[ V = V_{\text{cone}} + V_{\text{cylinder}} ]

Given that the volume of the solid is ( 72\pi ), we can write:

[ 72\pi = \frac{1}{3}\pi r^2 h_{\text{cone}} + \pi r^2 h_{\text{cylinder}} ]

Substituting the given values ( h_{\text{cone}} = 12 ) and ( h_{\text{cylinder}} = 28 ), we get:

[ 72\pi = \frac{1}{3}\pi r^2 (12) + \pi r^2 (28) ]

[ 72\pi = 4\pi r^2 + 28\pi r^2 ]

[ 72\pi = 32\pi r^2 ]

[ r^2 = \frac{72\pi}{32\pi} ]

[ r^2 = \frac{9}{4} ]

[ r = \frac{3}{2} ]

Now that we have the radius of the cylinder, we can find its area:

[ A_{\text{cylinder}} = \pi r^2 ]

[ A_{\text{cylinder}} = \pi \left(\frac{3}{2}\right)^2 ]

[ A_{\text{cylinder}} = \pi \times \frac{9}{4} ]

[ A_{\text{cylinder}} = \frac{9\pi}{4} ]

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Answer from HIX Tutor

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.

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.

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.

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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