2025 VJC P1 Q12

A chemical processing plant uses two types of automated dosing pumps, Pump \(A\) and Pump \(B\), to regulate the flow of a catalyst into a reaction chamber. Each pump is removed from the plant and tested over a \(10\)-hour trial period, and the volume of liquid dosed per hour is recorded.

The following data were collected:

Pump \(A\): The volume of liquid dosed was \(4.5\) litres in the first hour, and for each subsequent hour, the volume of liquid dosed decreased by a constant percentage of \(r\%\).
Pump \(B\): The volume of liquid dosed was \(4.7\) litres in the first hour, and for each subsequent hour, the volume of liquid dosed decreased by \(0.1\) litres.

1. Show that the total volume of liquid dosed by Pump \(A\) at the end of the trial period is \[\frac{450}{r} \left[ 1 - \left( 1 - \frac{r}{100} \right)^k \right] \text{ litres,}\] where \(k\) is a constant to be determined.[3]
2. It would be ideal if at the end of the trial period, the total volume of liquid dosed by Pump \(A\) is equal to that of Pump \(B\). Find the value of \(r\) for this ideal situation.[2]

After the trial period, the two pumps were cleaned and reset, and were subsequently installed in the actual plant. Assume that the two pumps perform consistently as what was observed during the trial period.

3. Pump \(A\) is installed for dosing in Reaction Line \(1\). To prevent underdosing, Pump \(A\) will cease to operate if the volume dosed in the next hour falls below \(4.0\) litres. With the assumption that \(r=1\), find the number of complete hours that Pump \(A\) should be in operation.[2]

Before operations begin in Reaction Line \(2\), Pump \(A\) breaks down.

4. To minimise disruption, the company considers buying a new motor for Pump \(A\) from a third-party supplier. The supplier claims that with the new motor, Pump \(A\) can dose at least \(360\) litres of liquid in total if it ran continuously without stopping. Determine the range of values of \(r\) for which the supplier's claim is invalid.[2]
5. The company eventually decides to use Pump \(B\) in Reaction Line \(2\). Using an algebraic method, determine if Pump \(B\) can dose at least \(150\) litres of liquid if left to run continuously without stopping.[3]