Answer:
<u><em>Dimension of box:-</em></u>
Side of square base = 10 in 
Height of box = 5 in 
Minimum Surface area, S = 300 in²
Step-by-step explanation:
An open box with a square base is to have a volume of 500 cubic inches.
Let side of the base be x and height of the box is y 
Volume of box = area of base × height 
                   
 
Therefore,  
 
It is open box. The surface area of box, S .
 
 
Put   
 

This would be rational function of surface area. 
For maximum/minimum to differentiate S(x) 
 
 
For critical point, S'(x)=0 
 
 
 
 
 
 
Put x = 10 into  
 
y = 5
Double derivative of S(x) 
 at x = 10
 at x = 10

Therefore, Surface is minimum at x = 10 inches
Minimum Surface area, S = 300 in²