If an open box has a square base and a volume of 115 in.^3 and is constructed fr
ID: 2877701 • Letter: I
Question
If an open box has a square base and a volume of 115 in.^3 and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. A rectangular box is to have a square base and a volume of 20 ft^3. If the material for the base costs $0.29 per square foot, the material for the sides costs $0.10 per square foot, and the material for the top costs $0.21 per square foot, determine the dimensions of the box that can be constructed at minimum cost.Explanation / Answer
V = L*W*H = 115 in3,
Since the base is square,
W = H.
L*W*W = 115,
L*W2 = 115,
To minimize the material,
let L = W
W*W*W = 115,
W3 = 115,
W = (115)1/3 = 4.863in.
L = 4.863in.
H = 4.863in.
b)
let the height be h cm
volume, V = x2 h = 20
h = 20 / x2
area of base = x2
area of top = x2
area of 4 sides = 4xh
Total Cost, C = 0.29x2 + 0.21 x2 + 0.1(4xh)
C = 0.5x2 + 0.4 xh
substitute h = 20 /x2
C = 0.5 x2 + 8 / x
differentiating
dC/dx = x - 8 x2
inorder to minimize cost , equate dC/dx to 0
x - 8/x2 = 0
x3 = 8
x = 2 ft
h = 20 /4 = 5
so dimensions of box are 2 x 2 x 5 ft