Problema Solution
a closed rectangular box is made from wood material.the length of the box will be twice its width and the box must hold a volume of 40 cubic feet. the wood material for the rectangular box is $4/sq. foot of surface area. find dimensions which will minimize the cost of the box and what the minimum cost will be.
Answer provided by our tutors
Let
x = the width of the box, x>0
2x = the length of the box
h = the height of the box, h>0
V = 40 ft^3 the volume of the box
V = width*length*height
x*2x*h = 40
h = 20/x^2
The surface area of the box is:
A = 2(x*2x + x*(20/x^2) + 2x*(20/x^2))
........
........
A = (1/4)(x^2 + 30/x)
We need to find the minimum of the function A = (1/4)(x^2 + 30/x)
the function has a minimum and we can see that from the graph:
A' = (1/4)(2x - 30/x^2)
(1/4)(2x - 30/x^2) = 0
........
........
x = 2.47 ft
2*2.47 = 4.94 ft
h = 20/(2.47)^2
h = 3.28 ft
The dimensions which will minimize the cost of the box are: the width is 2.47 ft, length is 2*2.47 = 4.94 ft and the height is 3.28.
The minimum cost will be:
4(1/4)(2.47^2 + 30/2.47)] = $18.25
The minimum cost will be $18.25.