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:  A(min)  =  2700Step-by-step explanation:Let   x   the side of the square base,   and h the height of the bin thenV(b)  = x²*h             ⇒  h  = 13500/x²      (1)Total area of the bin  = area of the base  + 4 sides each side x*hA(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/xTaking 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  =  0x³  -  27000 = 0x  =  30 inand   h  =  13500/x²          ⇒    h  = 13500/900h  =  15  inAnd finally the surface area is   A(min)  =  x²   +   54000/x        ⇒   A(min)  =  900  +  1800 A(min)  =  2700 in²