3.79: Cylinder with hemisphere: max volume

In this question all lengths are in centimetres. The volume of a cylinder with radius $r$ and height $h$ is $\pi r^{2} h$ and its curved surface area is $2\pi rh$. The volume of a sphere with radius $r$ is $\dfrac{4}{3}\pi r^{3}$ and its surface area is $4\pi r^{2}$.

The diagram shows a solid object in the shape of a cylinder of base radius $r$ and height $h$, with a hemisphere of radius $r$ on top. The total surface area of the object is $300\,\mathrm{cm}^{2}$.