A building has a leaking roof and, while it is raining, water drips into a 12 litre bucket - AQA - A-Level Maths Mechanics - Question 7 - 2021 - Paper 3

The expression ( W_n ) represents the volume of water dripping in the nth minute, which follows a geometric sequence due to the consistent reduction of 2% in volume each minute. Thus,

$W_n = A \times 0.98^{n-1}$

To find the value of A, we observe that in the first minute, the volume is 30 millilitres:

Hence, ( A = 30 ).

To determine the total increase in water after 15 minutes, we use the formula for the sum of the first n terms of the geometric series:

$S_{15} = \frac{30(1 - (0.98)^{15})}{1 - 0.98} \approx 392$

Thus, the increase in water in the bucket after 15 minutes is approximately 392 millilitres.

The maximum amount of water in the bucket can be calculated using the formula for the sum to infinity for a geometric series:

$S_\infty = \frac{30}{1 - 0.98} = 1500 \; \text{millilitres} (or 1.5 \; litres)$

Thus, the maximum amount of water in the bucket is 1.5 litres.

1. The model assumes that the water drips indefinitely, but in reality, the dripping might stop due to evaporation or blockages.
2. Some water may evaporate over time, reducing the total volume compared to the maximum calculated amount.