(a) It takes 8 students two- thirds of an hour to fill 12 tanks with water. How many tanks of water will 4 students fill in one- third of an hour at the sam...

Answer Details

(a) The number of tanks filled varies jointly with the number of students and the time worked: tanks \(= k \times (\text{students})\times(\text{time})\).

From the given data, 8 students in \(\tfrac{2}{3}\) h fill 12 tanks:

\[12 = k \times 8 \times \frac{2}{3} = \frac{16k}{3} \Rightarrow k = \frac{36}{16} = 2.25.\]

For 4 students in \(\tfrac{1}{3}\) h:

\[\text{tanks} = 2.25 \times 4 \times \frac{1}{3} = 2.25 \times \frac{4}{3} = 3.\]

So \(\mathbf{3}\) tanks are filled.

(b) A chord of length 20 cm is 12 cm from the centre. The perpendicular from the centre bisects the chord, giving a right triangle with legs 12 and 10.

Radius: \(r = \sqrt{12^2 + 10^2} = \sqrt{244} = 15.62\text{ cm}\).

(i) If \(\theta\) is the angle subtended at the centre, then in the right triangle \(\tan\left(\tfrac{\theta}{2}\right) = \dfrac{10}{12}\):

\[\frac{\theta}{2} = 39.8^\circ \Rightarrow \theta = 79.6^\circ.\]

(ii) The perimeter of the minor segment \(=\) chord \(+\) arc length. Arc length \(= \dfrac{\theta}{360^\circ}\times 2\pi r\):

\[\text{arc} = \frac{79.6}{360}\times 2(3.142)(15.62) = 0.2211 \times 98.16 = 21.7\text{ cm}.\]

\[\text{Perimeter} = 20 + 21.7 = 41.7\text{ cm}.\]