Homework SourseA rectangular storage container with an open top is to have
A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of its base is twice the width. Material for the base costs $13 per square meter. Material for the sides costs $10 per square meter. Find the cost of the materials for the cheaptest such container. Round the result to the nearest cent.
Solution
7) length is L width is W height is h L = 2W =======> the length of it\'s base is twice the width V= L * W * h V = 2W * W * h V = 2W^2 * h ======> h = 10 / 2W^2 ======> h = 5 / W^2 to find the cost function, the prices are dollars per m^2.......means that we need the surface area: S = W*L + [ Wh + Wh + Lh + Lh ] ( no top ) S = W*(2W) + 2Wh + 2Lh S = 2W^2 + 2Wh + 2(2W)h S = 2W^2 + 2Wh + 4Wh S = 2W^2 + 6Wh S = 2W^2 + 6W*(5/W^2) S = 2W^2 + 30/W ====> it is as S m^2 = 2W^2 m^2 + 30/W m^2 ====> times the prices C (x) = 2W^2 m^2 [ 13 $ / m^2 ] + 30/W m^2 [ 10 $ / m^2 ] C (x) = 26W^2 $ + 300/W $ C (x) = 26W^2 + 300/W ====> taking derivative C \' (x) = 52W + - 300 / W^2 =====> equal it to zero 0 = 52W + - 300 / W^2 52W = 300 / W^2 W^3 = 300/52 =======> W ˜ 1.794 m h = 5 / W^2 =====> h = 5 / (1.794)^2 ====> h ˜ 1.554 m L = 2W ====> L = 2 * 1.794 ˜ 3.587 m C (x) = 26W^2 + 300/W C (x) = 26(1.794)^2 + 300/(1.794) ˜ 250.90 $ ===========
