Question 818642
A storage bin consists of a cylindrical section for storing grain, with a conical roof.
 The height of the roof is one-third of the total height and the radius is 10 feet. 
The volume of the storage bin is 1000 cubic feet. Find the height.
:
Let x = the height of roof
then
2x = the height of the cylindrical section
and
3x = the total height
:
Cylinder Vol + Cone vol = 1000
{{{pi*10^2*2x}}} + {{{1/3}}}{{{pi*10^2*x}}} = 1000
{{{pi*200x}}} + {{{1/3}}}{{{pi*100x}}} = 1000
Divide thru by 100
{{{pi*2x}}} + {{{1/3}}}{{{pi*x}}} = 10
multiply by 3 to get rid of the fraction
{{{3(pi*2x)}}} + {{{pi*x}}} = 30
{{{6pi*x)}}} + {{{pi*x}}} = 30
combine like terms
{{{7pi*x}}} = 30
x = {{{30/(7pi)}}}
x = 1.364 ft is the height of the roof
3*1.364 ~4.1 ft is the height of the bin
:
This sounds kind of small, check it, 
{{{pi*10^2*2.73}}} = 857 cu/ft, cylindrical part
{{{1/3}}}{{{pi*n10^2*1.364}}} = 143 cu/ft cone part
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Total volume 1000 so guess it's right