# Multivariable optimisation for pipe insulation

I have two functions:

Heat Loss Cost $(C_h)$

$$C_h = 0.23 \times 8.67q$$

where $q$ is the heat loss of the fluid through an insulated pipe (in watts):

$$q = \frac{2 \pi L(T_1 - T_{air})}{\frac{ln(\frac{r_2}{r_1})}{K_p} + \frac{ln(\frac{r_3}{r_2})}{K_p}+ \frac{1}{r_3H_a}}$$

And annual cost of insulation $(C_i)$

$$C_i = L( 150\pi(r_3^2 - r_2^2) + 1 )$$

So that my total cost per year for piping a given fluid through the pipe $(C_t)$ is

$$C_t = C_h + C_i$$

What steps do I take to minimize the total cost, with respect to length of pipe $(L)$, inner radius of pipe $(r_1)$ outer radius of pipe $(r_2)$ and outer radius of pipe and insulation $(r_3)$ if $T_1$, $T_{air}$, $K_p$, $K_i$, $H_{air}$ are constants.

I am able to solve this with excel or like program however I need to be able to do it with calculus. I have tried partial differentiation but I am unsure what to set the partial derivatives to in order to find local min.

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## migrated from mathematica.stackexchange.comJun 6 '14 at 3:26

This question came from our site for users of Mathematica.

Sorry, but this site is for questions about the Mathematica software. Maybe you wanted mathematics.SE instead? –  Oleksandr R. Jun 6 '14 at 1:09