Volumes of Revolution (Edexcel A Level Further Maths: Core Pure)

Figure 1

A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student’s design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the -axis, the -axis, the line with equation  = 1, the curve and the curve  through 360° about the -axis.
The point A has coordinates (1, 0.5) and the point B has coordinates (0.5, 2.5) where the units are centimetres.

The curve  is modelled by the equation

Determine the value of and the value of  according to the model.

The curve  is modelled to be an arc of the circle with centre (0, 3).

Figure 1 shows the central vertical cross section  of a paddling pool that has a circular horizontal cross section. Measurements of the diameters of the top and bottom of the paddling pool have been taken in order to estimate the volume of water that the paddling pool can contain.

Using these measurements, the curve  is modelled by the equation

as shown in Figure 2.

Find the value of

The pool is being filled with water from a tap.

Find, in terms of , the volume of water in the pool when the pool is filled to a depth of .

Given that the pool is being filled at a constant rate of 15 litres every minute,

find, in , the rate at which the water level is rising in the pool when the depth of the water is .