Question

A company plans to manufacture a rectangular bin with a square base, an open top, and a volume of 13,500 in3. Determine the dimensions of the bin that will minimize the surface area. What is the minimum surface area? Enter only the minimum surface area, and do not include units in your answer.

Answer 1

Answer:  A(min)  =  2700

Step-by-step explanation:

Let   x   the side of the square base,   and h the height of the bin then

V(b)  = x²*h             ⇒  h  = 13500/x²      (1)

Total area of the bin  = area of the base  + 4 sides

each side x*h

A(b)  =  x²  +  4*x*h       ⇒

Area of the bin as fuction of x . From equation (1)

A(x)  =  x²  +  4*x*(13500/x²         A(x)  =  x²  + 54000/x

Taking derivatives both sides of the equation:

A´(x)   =  2*x  -   ( 54000/x²)

A´(x)  =  0        ⇒          2*x  -   ( 54000/x²)  =  0

( 2*x³   -54000)/x²                ⇒ 2*x³   - 54000  =  0

x³  -  27000 = 0

x  =  30 in

and   h  =  13500/x²          ⇒    h  = 13500/900

h  =  15  in

And finally the surface area is  

A(min)  =  x²   +   54000/x        ⇒   A(min)  =  900  +  1800

A(min)  =  2700 in²