Answer:
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 =
-----(1)
Now volume of the box V = a×a×b = a²b
From equation (1),
V = a²(
)
V = a(18 - 1.5a)
For greatest volume ,

![(d)/(da)[(18a-1.5a^(2) )]=0](https://img.qammunity.org/2021/formulas/mathematics/high-school/p3bomiigbj5h7r22moo96ga1alyoxch72k.png)
18 - 3a = 0
a = 6
Now form equation (1)
b =

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