Question

Optimization Calculus Problem

I understood that area= SA of hemisphere+ LA of Cyclinder and Volume is Volume of hemisphere + Volume of a Cyclinder

Cost equation is c(8) = 2pi radius squared plus 2pi radius times height

and Volume for h is (6000 divided by pi times radius squared )minus 2r divided by 3

My question is how did C = 12000 divided by r plus 44pi r squared divided by 3 come about? Or basically how did they get that C?

Answer 1

`\bf \begin{cases} V_c=\textit{cylinder's volume}\\ V_h=\textit{hemisphere's volume}\\ A_c=\textit{area of the cylinder}\\ A_h=\textit{area of the hemisphere} \end{cases} \\\\\\ V_c+V_h=6000\implies (\pi r^2h)+\left( (2\pi r^3)/(3) \right) \\\\\\ 6000=\cfrac{3\pi r^2h+2\pi r^3}{3}\implies 18000-2\pi r^3=3\pi r^2h \\\\\\ \boxed{\cfrac{6000}{\pi r^2}-\cfrac{2r}{3}=h} \\\\ -------------------------------\\\\`




`\bf A_c=2\pi rh+2\pi r^2\implies A_c=2\pi r\left(\cfrac{6000}{\pi r^2}-\cfrac{2r}{3} \right)+2\pi r^2 \\\\\\ A_c=\cfrac{12000}{r}-\cfrac{4\pi r^2}{3}\impliedby \textit{say it has a cost of 1, so stays the same}\\\\ -------------------------------\\\\ A_h=2\pi r^2\impliedby \textit{has a cost of 8 times the cylinder's} \\\\\\ A_h=16\pi r^2\\\\ -------------------------------\\\\`



`\bf \textit{total cost is }A_c+A_h \\\\\\ \cfrac{12000}{r}-\cfrac{4\pi r^2}{3}+16\pi r^2\implies \cfrac{12000}{r}-\cfrac{4\pi r^2}{3}+\cfrac{48\pi r^2}{3} \\\\\\ \boxed{C(r)=\cfrac{12000}{r}+\cfrac{44\pi r^2}{3}}`