Question

A closed box with a square base is to be built to house an ant colony. The bottom of the box and all four sides are to be made of material costing $1/ft², and the top is to be constructed of glass costing $5/ft². What are the dimensions of the box of greatest volume that can be constructed for $72?

Answer 1

Dimensions of the box will be 6 ft × 6 ft × 54 feet

Explanation:

A closed box is to be built having square base.

Let the dimensions of the box are a ft × a ft × b ft.

Material used to build the base and all four sides cost $1 per square ft.

Therefore, cost of base and four sides = 1[(a)² + 4ab]

Top of the box is to be constructed of glass costing $5 per square feet.

Cost of the top = 5a²

Total cost of the box material = (a² + 4ab) + 5a² = $72

6a² + 4ab = 72

b =
`(72-6a^(2))/(4a)` -----(1)

Now volume of the box V = a×a×b = a²b

From equation (1),

V = a²(
`(72-6a^(2))/(4a)`)

V = a(18 - 1.5a)

For greatest volume ,

`(dV)/(da)=0`

`(d)/(da)[(18a-1.5a^(2) )]=0`

18 - 3a = 0

a = 6

Now form equation (1)

b =
`18(6)-1.5(6)^(2)=108-54`

b = 54 feet.

Therefore, dimensions of the box will be 6 ft × 6 ft × 54 feet.