Question

The base and top are made from reinforced steel and the slides are made out of steel bars. If the total volume should be 1000 ft3, and the sides cost about 5 times less per unit area than the base, what dimensions will minimize the cost of the cage?

Answer 1

The dimensions that will minimize the cost of the cage are: base 5.85 x 5.85 ft and heght 29.2 ft.

Explanation:

We have a cage with dimensions x,y,z, with a fixed volume of 1000 ft3.

The sides cost 5 times less per unit of area than the base and top.

The volume can be written as:

`V=x\cdot y\cdot z=1000`

The cost function is

`C=top+base+sides=5xy+5xy+(2xz+2yz)`

The base will be square, so we can simplify as:

`V=x^2z=1000\\\\z=1000/x^2`

The cost become

`C=10xy+2(xz+yz)=10x^2+4xz=10x^2+4x((1000)/(x^2)) \\\\C=10x^2+4000/x`

To minimize the cost, we derive the cost function and equal to zero

`dC/dx=20x-4000x^(-2)=0\\\\20x=4000x^(-2)\\\\x^(1+2)=4000/20\\\\x^3=200\\\\x=\sqrt[3]{200}=5.85`

The base sides are 5.85 ft.

The height of the box (z) is:

`z=1000/x^2=1000/5.85^2=1000/34.2225=29.2`