- In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.}
The air pressure, \(P \mathrm {~kg} / \mathrm { cm } ^ { 2 }\), inside a car tyre, \(t\) minutes from the instant when the tyre developed a puncture is given by the equation
$$P = k + 1.4 \mathrm { e } ^ { - 0.5 t } \quad t \in \mathbb { R } \quad t \geqslant 0$$
where \(k\) is a constant.
Given that the initial air pressure inside the tyre was \(2.2 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\)
- state the value of \(k\).
From the instant when the tyre developed the puncture,
- find the time taken for the air pressure to fall to \(1 \mathrm {~kg} / \mathrm { cm } ^ { 2 }\)
Give your answer in minutes to one decimal place.
- Find the rate at which the air pressure in the tyre is decreasing exactly 2 minutes from the instant when the tyre developed the puncture.
Give your answer in \(\mathrm { kg } / \mathrm { cm } ^ { 2 }\) per minute to 3 significant figures.