N2022 P2 Q5

The diagram shows part of the circle \({{x}^{2}}+{{y}^{2}}={{r}^{2}}\) and the line \(y=r-h\), where \(0 < h < r\). The shaded region between the circle and the line is rotated about the \(y\)-axis to form a solid, which is called a spherical cap. The height of the spherical cap is \(h\).

1. Show by integration that the volume of the spherical cap is \(\frac{1}{3}\pi {{h}^{2}}\left( 3r-h \right)\).[5]
   
   An ornament is made from a solid sphere of radius \(15\) cm by removing two spherical caps, one of height \(p\) cm from the top of the sphere and the other height \(3p\) cm from the bottom of the sphere. The plane faces of the ornament are parallel (see diagram). The volume of the ornament, shown shaded, is \(3402\pi \) cm\(^{2}\).
   [It is given that the volume of a sphere of radius is \(\frac{4}{3}\pi {{r}^{3}}\).]
2. Find the cubic equation satisfied by \(p\) , and hence find the value of \(p\).[4]

A different ornament is made by making two parallel cuts to another sphere of radius \(15\) cm.

• The volume of this second ornament is less than the volume of the ornament in part (b).

• The top face of this second ornament has the same radius as the top face of the ornament in part (b).

• The bottom face of this second ornament has the same radius as the bottom face of the ornament in part (b).

3. Find the volume of this second ornament. Give your answer as an exact multiple of \(\pi \).[2]