10 A bottle of water has a temperature of \(6 ^ { \circ } \mathrm { C }\) when it is removed from a refrigerator.
It is placed in a room where the temperature is \(20 ^ { \circ } \mathrm { C }\)
10 minutes later, the temperature of the water is \(12 ^ { \circ } \mathrm { C }\)
The temperature of the water, \(T ^ { \circ } \mathrm { C }\), at time \(t\) minutes after it is removed from the refrigerator, may be modelled by the equation
$$T = 20 - a \mathrm { e } ^ { - k t }$$
10
- Find the value of \(a\).
10
- Calculate the value of \(k\), giving your answer to two significant figures.
10 - Using this model, estimate how long it takes the water to reach a temperature of
\(18 ^ { \circ } \mathrm { C }\) after it is taken out of the refrigerator. \(18 ^ { \circ } \mathrm { C }\) after it is taken out of the refrigerator.
10 - Explain why the model may not be appropriate to predict the temperature of the water three hours after it is taken out of the refrigerator.