Answer:
height = 14.5 ft 
width = length = 3.2 ft 
Step-by-step explanation:
Let the side of the square base is a and the height of the box is h. 
The material required to make the box is equal to the total surface area of the box. 
Area of base = a²
Cost of square base is $ 2 per ft²
So, the cost of making base = 2a²
Area of four walls + area of top = 2 (a² + ah + ah) + a² =  3a² + 2ah
Cost of making walls and the top is $ 1 per ft²
So, the cost of making walls and the top = 1 x ( 3a² + 2ah)
Total cost of making the box = 2a² + 3a² + 2ah = 5a² + 2ah
According to the question
144 = 5a² + 2ah 
 .... (1)
     .... (1) 
The volume of the box is V = length x width x height 
V = a x a x h 
V = 72 a - 2.5 a³     from equation (1)
Differentiate both sides with respect to a. 
dV/da = 72 - 7.5 a²
Put, it equal to zero for maxima and minima 
7.5 a² = 72 
a = 3.2 ft 
Put in equation (1)

h = 14.5 ft 
So, the dimensions of the box are 
height = 14.5 ft 
width = length = 3.2 ft