Gloria would like to construct a box with volume of exactly 45ft3 using only metal and wood. The metal costs $14/ft2 and the wood costs $5/ft2. If the wood is to go on the sides, the metal is to go on the top and bottom, and if the length of the base is to be 3 times the width of the base, find the dimensions of the box that will minimize the cost of construction. Round your answer to the nearest two decimal places.
w = width
L = length
h =height
w * L * h = 45
sides area = (2 w + 2 L)h
t and b area = 2 * L * w
==================
L = 3 w
so
3 w^2 h = 45 so h = 15/w^2
sides area = (2w+6w) h = 8 w (15/w^2) = 120/w
T and b area = 6 w^2
c = 5 *120/w + 14* 6 w^2
dc/dw = 0 at min = -600/w^2 + 168 w
w^3 = 600/168 = 3.57
w = 1.53
L = 4.59
h = 6.41