Optimization-Open Topped Cylinder

**Calculus23** · April 13th 2010, 04:45 PM

This question is really bothering me help would be much appreciated

The cost per square meter of the sides of an open-topped cylindrical tank is twice the cost per square meter of the bottom. Find the most economical proportions for a tank of given volume.

**skeeter** · April 13th 2010, 04:57 PM

Originally Posted by Calculus23

This question is really bothering me help would be much appreciated

The cost per square meter of the sides of an open-topped cylindrical tank is twice the cost per square meter of the bottom. Find the most economical proportions for a tank of given volume.

$V = \pi r^2 h$

$h = \frac{V}{\pi r^2}$

$C = \pi r^2 + 2(2\pi r h)$

sub in the expression for h in terms of r in the cost equation to get C as a function of r, simplify, and find dC/dr to find the minimum cost.

remember that V is just a constant.

**Calculus23** · April 13th 2010, 05:37 PM

I am still confused... any more help?

**skeeter** · April 13th 2010, 05:49 PM

Originally Posted by Calculus23

I am still confused... any more help?

I've told you exactly what to do.

1. substitute $\frac{V}{\pi r^2}$ in for $h$ in the cost equation.

2. simplify the cost equation.

3. take the derivative of the cost equation w/r to $r$ and minimize like you were taught in class.

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