2015 SRJC P2 Q2
A tank contains water which is heated by an electric water heater working under the action of a thermostat. When the water heater is first switched on, the temperature of the water is $35\text{ }{}^\circ \text{C}$. The heater causes the temperature to increase at a rate $r\text{ }{}^\circ \text{C}$ per minute, where $r$ is a constant, until the water temperature hits $75\text{ }{}^\circ \text{C}$. The heater then switches off.
(i)
Write down, in terms of $r$, the time taken for the temperature to increase from $\text{35 }{}^\circ \text{C}$ to $75\text{ }{}^\circ \text{C}$.
[1]
The temperature of the water then immediately starts to decrease. The temperature of the water at time $t$ minutes after the heater is switched off is $\theta \text{ }{}^\circ \text{C}$. It is known that the temperature of the water decreases at a variable rate $k\left( \theta -25 \right)\text{ }{}^\circ \text{C}$ per minute, where $k$ is a positive constant, until $\theta =35$.
(ii)
Write down a differential equation involving $\theta $ and $t$, to represent the situation as the temperature is decreasing.
[1]
(iii)
Given that when $\theta =55$, the temperature is decreasing at a rate of $5\text{ }{}^\circ \text{C}$ per minute, find the total length of time for the temperature to increase from $35\ {}^\circ \text{C}$ to $75\ {}^\circ \text{C}$ and then decrease to $35\ {}^\circ \text{C}$, leaving your answer in exact form, in terms of $r$.
[7]
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