Answer:
The dimensions of the box is 3 ft by 3 ft by 30.22 ft.
The length of one side of the base of the given box  is 3 ft.
The height of the box is 30.22 ft.
Step-by-step explanation:
Given that, a rectangular box with volume of 272 cubic ft.
Assume height of the box be h and the length of one side of the square base of the box is x.
Area of the base is = 
                                
The volume of the box  is = area of the base × height
                                            
Therefore,


The cost per square foot for bottom is 20 cent.
The cost to construct of the bottom of the box is
=area of the bottom ×20
  cents
 cents
The cost per square foot for top is 10 cent.
The cost to construct of the top of the box is
=area of the top ×10
  cents
 cents
The cost per square foot for side is 1.5 cent.
The cost to construct of the sides of the box is
=area of the side ×1.5
 cents
 cents
 cents
 cents
Total cost = 
                 
Let 
C
Putting the value of h


Differentiating with respect to x

Again differentiating with respect to x

Now set C'=0




Now 
Since at x=3 , C''>0. So at x=3, C has a minimum value.
The length of one side of the base of the box is 3 ft.
The height of the box is 
                                           =30.22 ft.
The dimensions of the box is 3 ft by 3 ft by 30.22 ft.