Question

We must build a box that has no top and whose base length is five times the base width. We have $1000.00 to buy materials to build this box. If the material for the sides costs $10.00 per square inch and the material for the bottom cost $15.00 per square inch determine the dimensions of the box that will have the greatest volume.

Answer 1

Surface area of open top box is given by:

S = L*W + 2*L*H + 2*W*H

Given that length is five times the base width, So L = 5*W

S = 5*W*W + 2*(5*W)*H + 2*W*H

S = 5W^2 + 12*W*H

Now given that we have total $1000 to buy material, And

Price of side materials = $10.00 inch^2

Price of bottom mateirals = $15.00 inch^2

So

P = Price = 15*(5*W^2) + 10*(12*W*H) = 1000

75W^2 + 120*W*H = 1000

H = (1000 - 75W^2)/(120W)

Now Volume of the box will be given by:

V = L*W*H

Since L = 5*W, So

V = 5*W*W*H = 5W^2*H

V = 5W^2*(1000 - 75W^2)/(120W)

V = (1/24)*(1000*W - 75*W^3)

Now Volume will be maximum at critical points, So for critical points

dV/dW = 0 = (1/24)*(1000*1 - 75*3*W^2)

1000 - 225*W^2 = 0

W^2 = 1000/225

W = sqrt (1000/225) = 2.108 inch

L = 5*W = 5*sqrt (1000/225) = 10.54 inch

H = (1000 - 75*(1000/225))/(120*sqrt (1000/225)) = 2.635 inch

So dimensions of box are: Length = 10.540 inch, width = 2.108 inch, Height = 2.635 inch

Let me know if you've any query.