The diagram is a portion of a right circular solid cylinder of radius 7 cm and height 15 cm. The centre of the base of the cylinder is Q, while that of the ...

From the diagram the solid is a slice (sector-prism) of a right circular cylinder: the sector angle at each centre is \(\angle ABC = \angle PQR = 120^\circ\), the radius is \(r = 7\text{ cm}\), and the height is \(h = 15\text{ cm}\). The solid is therefore \(\dfrac{120}{360} = \dfrac{1}{3}\) of the full cylinder.

(a) Volume

\[V = \frac{120}{360}\times \pi r^2 h = \frac{1}{3}\times \frac{22}{7}\times 7^2 \times 15\]

\[V = \frac{1}{3}\times \frac{22}{7}\times 49 \times 15 = \frac{1}{3}\times 22 \times 7 \times 15\]

\[V = \frac{1}{3}\times 2{,}310 = 770\text{ cm}^3\]

Correct to one decimal place, \(V = \mathbf{770.0\text{ cm}^3}\).

(b) Total surface area

The surface of the slice is made up of four parts:

1. Two sectoral faces (top \(ABC\) and bottom \(PQR\)), each a \(120^\circ\) sector of radius 7:

\[A_1 = 2\times \frac{120}{360}\pi r^2 = 2\times \frac{1}{3}\times \frac{22}{7}\times 49 = 2\times \frac{1}{3}\times 154 = 102.6667\text{ cm}^2\]

2. The curved outer face (a \(120^\circ\) strip of the cylinder wall):

\[A_2 = \frac{120}{360}\times 2\pi r h = \frac{1}{3}\times 2\times \frac{22}{7}\times 7 \times 15 = \frac{1}{3}\times 660 = 220\text{ cm}^2\]

3. The two flat rectangular faces formed by the straight cuts, each of dimensions \(r \times h = 7\times 15\):

\[A_3 = 2\times (7 \times 15) = 2\times 105 = 210\text{ cm}^2\]

Total surface area:

\[A = A_1 + A_2 + A_3 = 102.6667 + 220 + 210 = 532.6667\text{ cm}^2\]

Correct to one decimal place, \(A = \mathbf{532.7\text{ cm}^2}\).