A man is building a silo his back yard to store 1000 ft? of wintergreen lifesavers. Find the minimum amount of materials and the dimension of the silo to achieve it. Since his silo is a cylinder topped with a hemisphere, the formulas are as follows:V = π r? h + 2/3 π r?SA = 2 π r h + 2 π r?I don't know where to began. Can someone walk me through the steps to solve it (with solution)?
Use the volume equation to get an expression for h. 1000 = πr?(h + 2r/3) h = 1000/(πr?) - 2r/3 Substitute this expression for h into the surface area equation. A = 2πr[1000/(πr?) - 2r/3] + 2πr? A = 2000/r - 4πr?/3 + 2πr? A = 2000/r + 2πr?/3 Minimize A by taking derivative, setting it equal to zero, and solving for r. dA/dr = -2000/r? + 4πr/3 = 0 4πr/3 = 2000/r? r? = 1500/π r = (1500/π)^(1/3) ≈ 7.816 ft h = 1000/[π(7.816)?] - 2(7.816)/3 h = 0 (This means creating a silo shaped like a hemisphere uses less material than any silo with a cylindrical portion.) A = 2πr? A = 2π(7.816)? A ≈ 384 ft?
