5. A bath is filled with hot water. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of the water in the bath, \(t\) minutes after the bath has been filled, is given by
$$\theta = 20 + A \mathrm { e } ^ { - k t }$$
where \(A\) and \(k\) are positive constants.
Given that the temperature of the water in the bath is initially \(38 ^ { \circ } \mathrm { C }\),
- find the value of \(A\).
The temperature of the water in the bath 16 minutes after the bath has been filled is \(24.5 ^ { \circ } \mathrm { C }\).
- Show that \(k = \frac { 1 } { 8 } \ln 2\)
Using the values for \(k\) and \(A\),
- find the temperature of the water 40 minutes after the bath has been filled, giving your answer to 3 significant figures.
- Explain why the temperature of the water in the bath cannot fall to \(19 ^ { \circ } \mathrm { C }\).