Optimization Calculus Question Help (Thank You!!!)

Question

You have a budget of $1,200,000 to build a storage facility. The shape of this building must be a rectangular prism with the length double the width. If the floor costs $100/m², the walls cost $200/m², and the roof (which is flat) costs $500/m², what dimensions will create a storage facility with the largest volume?

Answer 1

Let x = width of base. Then 2x = length of base.

Let h = height

100(2x²) + 500(2x²) + 200(4xh + 2xh) = 1,200,000

1200x² + 1200xh = 1,200,000

x² + xh = 1000

So, h = (1000 - x²) / x

Volume = V = x(2x)[(1000 - x²)/x] = 2000x - 2x³

dV/dx = 2000 - 6x² = 0

x = width = (10/3)√30

2x = length = (20/3)√30

height = h = (1000 - x²) / x = (20/3)√30

Answer 2

walls, 2Lh+2Lw, cost 200 per m^2

floor, Lw, cost 100 per m^2

roof, Lw, cost 500 per m^2

length = twice width L=2w

budget allows only $1,200,000

what is the rectangular prism with largest volume not over budget?

here's the solution to a different problem, same but with w=L

dimensions are:

height=30sqr(5/3) = about 38.73 meters

length & width each 20sqr(5/3) = about 25.82 meters

max volume = 20,000sqr(5/3) = 25,819.89 cubic meters

largest would be a cube, with L=w=h, but not cheapest given different prices for the sides

V=Lwh= hw^2

Total Cost = 1,200,000 =600w^2 + 800wh

solve for h

substiute into V

take V' and set = 0, solve for w=L

back to the original posted problem with L=2w

then

dimensions are

width = 20 meters

Length = 40 meters

height = 30 meters

Volume = 24,000 cubic meters

Total Cost = $1,200,000

Total Cost = cost of floor + cost of roof + cost of 4 sides

= 100(20)(40) + 500(20)(40) + 2(200)(20)(30) +2(200)(40)(30)

= 480,000 + 720,000

=1,200,000