A farmer's silo is the shape of a cylinder with a hemisphere as the roof. If the height of the silo is 78 feet and the radius of the hemisphere is r feet, express the volume of the silo as a function of r.
total ht. of silo = 78 ft, so ht. of cylindrical portion is (78-r) ft vol. of cylindrical portion = π r ?(78 - r) vol. of of hemispspherical part = 2/3π r ? so vol. of silo = 2π r ?(39 - r/2 + r/3) = 2π r ?(39 - r/6) -------------------------
total volume = volume of cylinder + volume of hemisphere =1/3(pi*r^2*h) + (2/3)pi*r^3 here the radii of the hemisphere and the cylinder are equal. also 78 = h+r ie, h = 78-r. using this in the above equation,we get the volume in terms of r. total volume = (1/3)(pi*r^2)(78-r)+(2/3)pi*r^3 you can further simplify it.
The silo is composed of a cylinder and a hemisphere. The total volume of the silo = vol. of cylinder + vol. of hemisphere The volume of a cylinder is given by pi*r^(2)*H where r is the radius of the cylinder and H is the height of the cylinder. For the silo's cylinder r = r and H = 78 - r. Hence its volume = pi r^(2)*(78 - r) The volume of a sphere of radius r is (4/3)pi*r^(3). Hence the volume of a hemisphere is (2/3)pi*r^(3). Hence the volume of the silo = pi r^(2) (78 - r) + (2/3)pi*r^(3) = pi*r^(2) [ 78 - r/3]
