Question

Really need help with this, am I on the right track or no?

Answer 1

`\bf a)\\\\ V=\stackrel{\textit{volume of cone}}{\cfrac{\pi r^2\cdot \boxed{3r}}{3}}+\stackrel{\textit{volume of hemisphere}}{\cfrac{2\pi r^3}{3}}\implies V=\cfrac{3\pi r^3}{3}+\cfrac{2\pi r^3}{3} \\\\\\V=\cfrac{5\pi r^3}{3} \\\\\\ S=\stackrel{\textit{lateral area of cone}}{\pi r\sqrt{r^2+\boxed{3^2r^2}}}+\stackrel{\textit{area of hemisphere}}{2\pi r^2}\implies S=\pi r√(r^2+9r^2)+2\pi r^2`


`\bf S=\pi r√(10r^2)+2\pi r^2\implies S=\pi r^2√(10)+2\pi r^2 \\\\\\ S=\pi r^2(2+√(10))\\\\\\ b)\\\\ \stackrel{A=kS}{A=k\cdot \pi r^2(2+√(10))}\qquad \qquad \stackrel{C=kV}{C=k\cdot \cfrac{5\pi r^3}{3}}`


`\bf c)\\\\ A\ge C\implies k\cdot \pi r^2(2+√(10))\ge k\cdot \cfrac{5\pi r^3}{3} \\\\\\ k\pi r^2(2+√(10))\ge \cfrac{5k\pi r^2r}{3}\implies \cfrac{\underline{k\pi r^2}(2+√(10))}{\underline{k\pi r^2}}\ge\cfrac{5r}{3} \\\\\\ 3(2+√(10))\ge 5r\implies 6+3√(10)\ge 5r\implies \cfrac{6+3√(10)}{5}\ge r`