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(A) Let $r$ be the common radius and $h$ be the height of the cylinder.The total surface area $S$ of the solid is the sum of the curved surface area o

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$1: 1$

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(A) Let $r$ be the common radius and $h$ be the height of the cylinder.The total surface area $S$ of the solid is the sum of the curved surface area of the hemisphere $(2\pi r^2)$,the curved surface area of the cylinder $(2\pi rh)$,and the base area of the cylinder $(\pi r^2)$.$S = 2\pi r^2 + 2\pi rh + \pi r^2 = 3\pi r^2 + 2\pi rh$From this,we can express $h$ in terms of $S$ and $r$:$h = \frac{S - 3\pi r^2}{2\pi r}$The volume $V$ of the solid is the sum of the volume of the hemisphere $(\frac{2}{3}\pi r^3)$ and the volume of the cylinder $(\pi r^2 h)$:$V = \pi r^2 h + \frac{2}{3}\pi r^3$Substituting the expression for $h$:$V = \pi r^2 \left( \frac{S - 3\pi r^2}{2\pi r} \right) + \frac{2}{3}\pi r^3$$V = \frac{1}{2} (Sr - 3\pi r^3) + \frac{2}{3}\pi r^3 = \frac{1}{2} Sr - \frac{3}{2}\pi r^3 + \frac{2}{3}\pi r^3 = \frac{1}{2} Sr - \frac{5}{6}\pi r^3$To find the maximum volume,differentiate $V$ with respect to $r$ and set it to $0$:$\frac{dV}{dr} = \frac{S}{2} - \frac{5}{2}\pi r^2 = 0$$S = 5\pi r^2$Now,substitute $S = 5\pi r^2$ back into the expression for $h$:$h = \frac{5\pi r^2 - 3\pi r^2}{2\pi r} = \frac{2\pi r^2}{2\pi r} = r$Thus,the ratio of the height of the cylinder to the common radius is $h:r = 1:1$.

$A$ solid hemisphere is mounted on a solid cylinder,both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume,then the ratio of the height of the cylinder to the common radius is

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