86) square based prism

Question

1) a square based prism must have a volume of 8000 cm cube. what dimensions should be used to maximize the surface area?
 explain please 

i dont understand how to do this question below. explain. 

2) a cereal box has dimensions of 10 cm x 24 cm x 33 cm.

what is the volume of the cereal box?

 b) these dimensions does not make the surface area a minimum. suggest one reason why the manufacturer would not use dimensions that minimize the surface area 

Answer 1

Problem 1 is an optimization problem in calculus. 

Finding the maximum surface area is a degenerate problem because as the prism is made very tall with small base, the surface area grows without limit.   The more interesting problem is to find the dimensions which minimize the area.  This analysis is given below.

 

  The volume of a square base prism is   V =  h s²  ,  where h is the height and s is the length of the edge of the square.    Since V = 8000,     h =  8000/s²  .

 

The area is A =  4 s h  +  2 s²     substituting for h  gives

 

    A =   4 s 8000/ s² + 2 s²  =   32000/s + 2s² 

  To find the minimum area, differentiate A with respect to s and set equal to zero.  This procedure results in

   0 =   -32000/s² + 4 s      Solving for s gives s =  20 cm.    So h = 20 cm and   A = 2400 cm²

  Thus the minimum area prism is a cube. 

 

For problem 2,   the optimal box shape would be a cube  (sphere would be even better!).   Grocers like packages that have one relatively thin dimension so that they can be put on the shelf one layer deep.