I am working on the solution of the following problem.
A cylinder has a flat base on one end, and a hemispherical top on the other. The material used for the hemisphere is twice the cost of the material used for the cylinder per m$^2$. Find the most economic proportions, given that the object is hollow.
I constructed a formula for the cost based on the diagram below:
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The total cost is proportional to the total materials used; i.e. $$\mathrm{Cost}\propto \mathrm{Base~Area}~+~\mathrm{Cylinder~Surface~Area}~+~2~\times~\mathrm{Surface~Area~of~Hemisphere}$$ $$\Rightarrow \mathrm{ Cost }=k(\pi r^2+2\pi r(h-r)+2(2\pi r^2))~~~\text{for some }k\in\mathbb R$$ So solving $\displaystyle\frac{\partial}{\partial r}(\text{Cost})=0~$ should give the optimal proportions. The only issue is that this gives the result $$r=-\frac{h}{3}$$
Which doesn't really make sense to me. How can the radius of the base/hemisphere be a negative third of the height? Can anyone help me find the flaw in my reasoning?
You need to assume that the volume is fixed so that you get $h$ as a function of $r$ which can be replaced in the expression for the cost. Then solve $\frac {dC}{dr}=0$