A silo(base not included) is to be constructed in teh form of a cylinder with a hemisphere on top. The cost of the construction per square unit of surface area is twice as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed and the cost of the construction is to be kept at a minimum. Neglect the thickness of the Silo and the waste in construction. I got r=3v/8pi to the 1/3
The volume of the hemisphere is 2/3*pi*r^3, and the volume of the cylinder is pi*r^2*h. 2/3*pi*r^3+pi*r^2*h=V. pi*r^2*h=V-2/3*pi*r^3. h=V/(pi*r^2)-2/3*r. The lateral surface area of the hemisphere is 2*pi*r^2, and the lateral surface area of the cylinder is 2*pi*r*h. A=2*pi*r^2+2*pi*r*h. A=2*pi*r^2+2*pi*r(V/(pi*r^2)-2/3*r). A=2*pi*r^2+2*V/r-4/3*pi*r^2. dA/dr=4*pi*r-2*V/r^2-8/3*pi*r. dA/dr=4/3*pi*r-2*V/r^2. 4/3*pi*r-2*V/r^2=0. 4/3*pi*r=2*V/r^2. 4/3*pi*r^3=2*V. r^3=3/2*V/pi. r=(3V/(2pi))^(1/3). I'm too lazy to figure out the height in terms of V. It would be really nice if they specified what the fixed volume was!
