SOLUTION: A storage bin for corn consists of a cylindrical section made of wire mesh, surmounted by a conical tin roof, as shown in the figure. The height of the roof is one-third the height

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Question 120235This question is from textbook College Algebra
: A storage bin for corn consists of a cylindrical section made of wire mesh, surmounted by a conical tin roof, as shown in the figure. The height of the roof is one-third the height of the entire structure. If the total volume of the structure is 2600 ft3 and its radius is 9 ft, what is its height? (Round the answer to one decimal place.)
Thank you for your help!!! This question is from textbook College Algebra

Answer by solver91311(24713) About Me  (Show Source):
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You have two solid figures whose volume adds up to 2600ft%5E3

let the height of the cylinder part be h, then the height of the conical part is h/3.

The volume of the cylinder is V%5Bcyl%5D=pi%2Ar%5E2%2Ah, and the volume of the conical part, in terms of the height of the cylinder is V%5Bcone%5D=%281%2F3%29%2Api%2Ar%5E2%2A%28h%2F3%29

Add these two volumes together V=%28pi%2A9%5E2%2Ah%29%2B%28%281%2F3%29%2Api%2A9%5E2%28h%2F3%29%29=2600ft%5E3

Simplify:

%28pi%2A9%5E2%2Ah%29%2B%28pi%2A9%5E2%28h%2F9%29%29=2600ft%5E3

%28pi%2A9%5E2%29%28h%2B%28h%2F9%29%29=2600ft%5E3

%28pi%2A9%5E2%29%2810h%2F9%29=2600ft%5E3

%28pi%2A9%29%2810h%29=2600ft%5E3

h=%28%282600%29%2F%2890%2Api%29%29ft

h=%28%28260%29%2F%289%2Api%29%29ft which is the height of the cylinder part.

To that we have to add the height of the conical part, h%2F3=%28%28%28260%29%2F%289%2Api%29%29%2F3%29ft=%28%28260%29%2F%2827%2Api%29%29ft

So the total height is .

A little calculator work shows this to be approximately 12.26. Your question requires rounding to one decimal place, so you want 12.3 -- however, it is really inappropriate to express this answer to the nearest 10th, since the given volume was expressed only to the nearest whole cubic foot. 12.3 feet would be correct IF the volume had been given as 2600.0 cubic feet. You should never express an answer with greater precision than the least precise of your given measurements.

Hope this helps,
John