Optimize the cost of steel $S$

Question

$$S= 10 \ell^{0.4} C^{0.6}$$ $$\text{Total Cost}= 60\ell+100C$$ $$40000= 10 \ell^{0.4} C^{0.6}$$ $S$ is the function of $\ell$ and $C$? We have to minimize the total cost How to proceed in this problem?? I know I'll have to minimize the cost for that I'll have to minimize l and C which are variables here

Answer 1

Hint

From the constraint $$40000=10 \ell^{2/5}\,C^{3/5}$$ you can extract $$C=\frac{800000 \sqrt[3]{2}}{\ell^{2/3}}$$ which makes $$\text{Total Cost}=60 \ell + \frac{80000000 \sqrt[3]{2}}{\ell^{2/3}}$$ what you want to minimize.

I am sure that you can take it from here.