Question

A company wishes to manufacture a box with a volume of 2424 cubic feet that is open on top and is twice as long as it is wide. find the width of the box that can be produced using the minimum amount of material

Answer 1

Let x = the width
Let 2x = the length
Let h = the height
then vol = x*2x*h. So we have 2x^2*h = 24
h=24/(2*x^2)=12/x^2
Surface area: two ends + 1 bottom + 2 sides (no top)
S.A. = 2(x*h) + 1(2x*x) + 2(2x*h)
S.A. = 2xh + 2x^2 + 4xh S.A. = 2x^2 + 6xh
Replace h with 12/x^2
S.A = 2x^2 + 6x(12/x^2)
S.A = 2x^2 + 6(12/x)
S.A = 2x^2 + (72/x)
Graph this equation to find the value of x for minimum material
Min surface area when x = 3.0 is the width then
2(3) = 6 is the length
Find the height:
h=12/(3.0)^2
h=1.33
Box dimensions for min surface area: 3.0 by 6 by 1.33; much better numbers
Check the vol of these dimensions: 3.0*6*1.33 ~ 24
graphic attachment