Closed box rectangle

Question

a rectangular closed box of height h with a square base of side b has a volume V = 64m3
. the two
squares bottom and the top side are made of material cost $20/m2 and the remaining sides cost
$10/m2
. Find b and h for which it minimize the cost of the box? (be sure to show me that is the
minimum)

Answer 1

My calculus is a bit rusty but here is how I solved this problem.

 

Write out your information

Volume=b²h=64

Cost=20(b²)+10(bh)

substitute in h=64/b²

C=20b²+640/b

change this to C=y and b=x

y=20x²+640/x

 

now you can graph this equation and find the minimum when x>0 

you can also take a derivative and prove that this is the lowest cost of the box.

 

The following link has a good explanation of similar problems:

http://tutorial.math.lamar.edu/Classes/CalcI/Optimization.aspx