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

Answer:

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

Step-by-step 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 = $\frac{72-6a^{2}}{4a}$ -----(1)

Now volume of the box V = a×a×b = a²b

From equation (1),

V = a²($\frac{72-6a^{2}}{4a}$)

V = a(18 - 1.5a)

For greatest volume ,

$\frac{dV}{da}=0$

$\frac{d}{da}[(18a-1.5a^{2} )]=0$

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.