Two pumps, A and B, are capable of filling a liquid reservoir in 25 minutes and 30 minutes respectively. A third pump, C, can fill the reservoir at a rate of 15 litres per minute. If all three pumps function simultaneously to fill the reservoir in 10 minutes, determine the total capacity of the reservoir (in litres).

The correct answer is 562.5 litres.

Let us break down the problem step-by-step to find the total capacity of the reservoir.

Let the total capacity of the reservoir be $V$ litres.

First, let us calculate the rate at which each pump fills the reservoir per minute:

1. Work rate of Pump A:
Pump A can fill the entire reservoir of capacity $V$ in 25 minutes.
Rate of Pump A = $\frac{V}{25}$ litres per minute.

2. Work rate of Pump B:
Pump B can fill the entire reservoir of capacity $V$ in 30 minutes.
Rate of Pump B = $\frac{V}{30}$ litres per minute.

3. Work rate of Pump C:
Pump C fills the reservoir at a given rate of 15 litres per minute.

When all three pumps function simultaneously, their combined filling rate per minute is the sum of their individual rates:
Combined Rate = Rate of A + Rate of B + Rate of C

$\text{Combined Rate}=\frac{V}{25}+\frac{V}{30}+15$

We are given that all three pumps working together fill the entire reservoir in 10 minutes.
Therefore, the total volume filled in 10 minutes is equal to $V$:

$10\times \left(\frac{V}{25}+\frac{V}{30}+15\right)=V$

Now, let us solve this equation for $V$:

Divide both sides of the equation by 10:

$\frac{V}{25}+\frac{\mathrm{V<mn>30</mn>}}{}+15=\frac{V}{10}$

Rearrange the terms to group all terms containing $V$ on one side:

$15=\frac{V}{10}-\frac{V}{25}-\frac{V}{30}$

To combine the fractions on the right side, find the least common multiple (LCM) of the denominators 10, 25, and 30, which is 150:

$15=\frac{15V-6V-5V}{150}$

Simplify the numerator:

$15=\frac{4V}{150}$

$15=\frac{2V}{75}$

Now, solve for $V$:

$2\mathrm{V<mo>=</mo><mn>15</mn><mo>\times </mo><mn>75</mn>}$

$2\mathrm{V<mo>=</mo><mn>1125</mn>}$

$V=\frac{1125}{2}=562.5$

Thus, the total capacity of the reservoir is 562.5 litres.