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

The amount of water dripped each minute forms a geometric sequence where the first term is $30$ millilitres and each subsequent term is reduced by a factor of $0.98$. Hence:

$W_n = 30 \times 0.98^{n-1}$

In this case, $A = 30$.

First, we need to find the total amount of water that drips into the bucket in 15 minutes:

$S_{15} = 30 + 29.4 + 28.8 + ... + W_{15}$

Applying the formula for the sum of a geometric series:

$S_n = A \frac{1 - r^n}{1 - r}$

Where $A = 30$, $r = 0.98$, and $n = 15$, the total increase is approximately:

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

Thus, the increase in the water in the bucket is $392$ millilitres.

The maximum amount of water occurs when it stops dripping, found using the formula for the sum to infinity:

$S_\infty = \frac{A}{1 - r} = \frac{30}{1 - 0.98} = 1500$

Thus, the maximum amount of water in the bucket is $1500$ millilitres.

1. The model assumes that water continues to drip indefinitely, which is unrealistic as the dripping stops eventually.

2. Environmental factors such as evaporation may also affect the total amount of water collected over time.