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 ft?. The quot;hemisphere attached to the topquot; part is giving me problems.
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.
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
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.
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)
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.
