Expert: Alon Mandes - 5/11/2009

Question
a manufacturer wants to create closed rectangular box with a square base. it costs $2 per square foot to create the base of this box, ans $1 per square foot for the rest of the box. if the box must hold 2000 cube feet in volume, then
a.) determine the dimensions that minimize the cost of the box.
b.) determine the minimum cost

for the answer, i got lenght=10, width, because its equal to length, as it is a square base= 10 and height=20. with a minimum cost of $1200, i did everything right and still the professor counted it wrong, if it is wrong can you shome me the prosses to get it right. thanks

Answer
If we consider x as the dimensions of the base & y as the height,
then the surface area of the box will be S=2x²+4xy. We also know
that the volume is 2000, thus x²y=2000 -> y=2000/x² .
Now, the coast will be C=2$*2x²+1$*xy=4x²+2xy $.
C(x)=4x²+2x(2000/x²)=4x²+4000/x . Now we derive C & set it equal 0:
C'(x)=8x-4000/x²
C'(x)=0 -> 8x=4000/x² -> x³=500 -> x=7.93 , now we calculate y :
y=2000/x²=2000/62.99=31.74
The total coast will be : 4x²+2xy $ = 4(7.93)²+2(7.93)(31.74) $
=754.936 $

Alon.