# [SOLVED] optimization problem (cost function)

• May 3rd 2009, 04:27 PM
coolguy00777
[SOLVED] optimization problem (cost function)
I can find the cost function, but I don't know how to find minimum cost. Which should = $329.34 A rectangular storage container with an open top is to have a volume of 20 m^3. The length of its base is twice the width. Material for the base costs$15 per square meter.Material for the sides costs $7 per square meter. 1. Find the cost function for the container. 2. Find the cost of materials for the cheapest such container.$\displaystyle
V=20=2x^2y
\displaystyle
Surface Area = 2x^2+6xy
$Cost Function:$\displaystyle
15*(2x^2) + \frac{7*60}{x}
\displaystyle
30x^2 + \frac{420}{x}
$• May 3rd 2009, 05:02 PM SengNee Quote: Originally Posted by coolguy00777 I can find the cost function, but I don't know how to find minimum cost. Which should =$329.34

A rectangular storage container with an open top is to have a volume of 20 m^3. The length of its base is twice the width. Material for the base costs $15 per square meter.Material for the sides costs$7 per square meter.
1. Find the cost function for the container.
2. Find the cost of materials for the cheapest such container.

$\displaystyle V=20=2x^2y$
$\displaystyle Surface Area = 2x^2+6xy$
Cost Function:
$\displaystyle 15*(2x^2) + \frac{7*60}{x}$
$\displaystyle 30x^2 + \frac{420}{x}$

To find the max or min,you just find the stationary point of the graph and identify its concavity.

$\displaystyle \frac{dy}{dx}=0$
$\displaystyle \frac{d^2y}{dx^2}$ is positive or megative.
• May 4th 2009, 07:45 AM
coolguy00777
sorry, but I'm still lost. Is there not an algebraic way to find the answer? What part do I take the derivative of?
• May 4th 2009, 07:57 AM
skeeter
Quote:

Originally Posted by coolguy00777
sorry, but I'm still lost. Is there not an algebraic way to find the answer? What part do I take the derivative of?

your cost function ...

$\displaystyle \frac{d}{dx}\left[C = 30x^2 + \frac{420}{x}\right]$

$\displaystyle \frac{dC}{dx} = 60x - \frac{420}{x^2} = 0$

$\displaystyle x = \sqrt[3]{7}$

$\displaystyle C(\sqrt[3]{7}) = 329.34$