2016 AJC Promo Q10

[It is given that a sphere of radius \(r\) has volume \(\frac{4}{3}\pi {{r}^{3}}\).]

A toy is made up of two parts.

• A hemispherical top of radius \(r\) cm.

• A cylindrical base of radius \(r\) cm and height \(h\) cm.

The diagram shows the toy enclosed in a transparent right conical container of radius \(5\) cm and height \(12\) cm, with its base in contact with the circular base of the container, and its hemispherical top in contact with the inner surface of the container.

Given that \(r\) and \(h\) vary,

1. show that \(h=12-\frac{13}{5}r\) ;[2]
2. using differentiation, find the values of \(r\) and \(h\) such that the toy has a maximum volume;[7]
3. sketch the graph showing the volume of the toy as the radius of the hemisphere varies.[3]