Algebra word problem?
A large grain silo is to be constructed in the shape of a circular cylinder with a hemisphere attached to the top. The diameter of the silo is to be 30 feet, but the height is yet to be determined. Find the height h of the silo that will result in a capacity of 11,250π ft³.
The "hemisphere attached to the top" part is giving me problems.
Comments
the diameter and radius for both the cylinder as well as hemisphere is the same.
volume of hemisphere=2/3*p* r^3
volume of cylinder=p*r^2*h(p=pie)
total volume=11250p=2/3pr^3+p*r^2*h
11250=(2/3r+h)r^2
h=40m
The given silo structure has a cylinder on the top hemispher is placed. --- Imaging a hollow straw close at one end
a sample picture searched from net --> http://www.easymaths.com/tank.gif
so the diameter of the cylinder is equatl to the diameter of the hemisphere.
so the total volume of the whole structure is
= Volume of cylinder + volume of hemi sphere
= pi * r * r * h + 2/3 * pi * r * r * r
= pi * r * r (h + 2/3r) = 11,250 pi
so r* r ( h + 2/3 r) = 11,250 ( r is 15 feet)
so h is 40 ft
so the height of the silo is height of the cylinder + radius of the hemisphere which is 55 feet.
Volume of the cylinder is V=pi*r^2*height
Volume of the hemisphere is 4/3 * pi * r^3
So total volume (V) is pi*r^2*height (H) + 4/3 * pi * r^3
or
V=pi*r^2*H+4/3*pi*r^3
V=pi*r^2*(H+4/3*r)
V/(pi*r^2)=H+4/3*r
H=V/(pi*r^2) - 4/3*r
Substituting 15 for r and 11250 pi for V,
H=((11250*pi)/(225*pi)) - 20
H=50-20
H=30
The cylinder is 30 feet high. Because it is also 30 feet wide, a hemisphere on top will add an additional 15 feet, and the topmost point of the silo is 45 feet above the base.
Edit: I used the formula for the volume of a sphere, not of a hemisphere. So my answer will be wrong. Ignore it.
Vol. of hemisphere = 2/3 pi r^3
= 2/3 pi * 3375 cu ft = 2250 pi ft^3
Therefore vol of cylindrical part = (11250 - 2250) pi ft^3
= 9000 pi ft^3
Vol of cylinder = pi r^2 *h
= 225 pi * h
Therefore 225 pi * h = 9000 pi
therefore 225 h = 9000 (Dividing both sides by pi)
therefore h = 40 (This is height of cylinder)
Therefore total height = 55 ft (height of cylinder + hemisphere)
Okay, volume for half a sphere is (.5)(4/3)pi*r^3. SO, the hemisphere is
.5(4/3)pi*(30/2)^3=2250pi.
SO, the height is (11,250pi-2250)/(pi*(30/2)^2) = 40.
SO, it should be 40 feet.
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