1. ## Optimization: Minimization

Hey everyone, here is a minimization problem from the Optimization II chapter in my book. I have the maximization down, but this I can't seem to get. Can someone work through this one for me and show the steps so I can see the flow?

Minimizing Packagin Costs.
A rectangular box is to have a square base and a volume of 20 ft cubed. If the material for the base costs 30 cents/square foot, the material for the sides costs 10 cents/square foot, and the material for the top costs 20 cents/square foot, determine the dimensions of the box that can be constructed at minimum cost.

2. Volume is given by:

$\displaystyle x^{2}y=20$.........[1]

Cost with regard to surface area is:

$\displaystyle 0.30x^{2}+0.40xy+0.20x^{2}$..........[2]

Solve [1] for y and sub into [2]:

$\displaystyle 0.30x^{2}+0.40x(\frac{20}{x^{2}})+0.20x^{2}=0.5x^{ 2}+\frac{8}{x}$

Diferentiate, set to 0 and solve for x. y will follow.

3. Hello, 2taall!

We use the same procedure of Minimization as for Maximization.
The tricky part (of course) is setting up the function to optimize.
I'll walk you thorough the set-up.

A rectangular box is to have a square base and a volume of 20 ft³.
The material for the base costs 30˘/ft², for the sides 10˘/ft², and the top 20˘/ft².
Determine the dimensions of the box that can be constructed at minimum cost.
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x
The volume of the box is: .$\displaystyle x^2y \:=\:20\quad\Rightarrow\quad y \:=\:\frac{20}{x^2}$ .[1]

The base has area $\displaystyle x^2$ ft². .Its cost is: .$\displaystyle 30x^2$ cents.

The top has area $\displaystyle x^2$ ft². .Its cost is: .$\displaystyle 20x^2$ cents.

The sides have area $\displaystyle 4xy$ ft². .Their cost is: .$\displaystyle 40xy$ cents.

Hence, the total cost is: .$\displaystyle C \;=\;30x^2 + 20x^2 + 40xy\quad\Rightarrow\quad C \:=\:50x^2 + 40xy$ .[2]

Substitute [1] into [2]: .$\displaystyle C \:=\:50x^2 + 40x\left(\frac{20}{x}\right)$

Hence, we must minimize: .$\displaystyle C \;=\;50x^2 + 800x^{-1}$

4. Ok guys,

I differentiated .50x^2 + 8/x

I got f'x= x - (8/ x^2 )=

solving for x:
-8/x^2 = -x =

-8= -x^3 =

8=x^3 =

taking the cubed root of each side:
x=2

Can you guys check that math?

Now we have(hopefully) x=2 and y= 20/x^2

EDIT: is the x=2 plugged back into the (x^2)y=20?
in which case [(2)^2]y=20 would make y=5
so the box would be 5 x 2 x 2 ?

Thanks everyone.

5. I appreciate all the help you guys are giving. I only wish that I had found this forum at the beginning of my Calculus course!

Was everything OK as far as my work went?

Thanks.

6. Hello,2taal!