Arectangular box with square base and open at the top is to have a capacity of 16,823 cu. cm. find the height of the box that requires minimum amount of the material required.

a. 14.12cm

b. 16.14cm

c. 12.13cm

d. 10.36cm

let the side of base =x and height=h
 
so
 
surface area of the box = S(x)= x² +4xh
 
to find the minimum we calculate the first derivative
S'(x) = 2x + 4h
this is minimum for S'(x) =0
 
2x + 4h = 0 replace x by its value..

square both sides
 
cross multiply
 

 
so