10 As part of an experiment, Zena puts a bucket of hot water outside on a day when the outside temperature is \(0 ^ { \circ } \mathrm { C }\).
She measures the temperature of the water after 10 minutes and after 20 minutes. Her results are shown below.
| Time (minutes) | 10 | 20 |
| Temperature (degrees Celsius) | 30 | 12 |
Zena models the relationship between \(\theta\), the temperature of the water in \({ } ^ { \circ } \mathrm { C }\), and \(t\), the time in minutes, by
$$\theta = A \times 10 ^ { - k t }$$
where \(A\) and \(k\) are constants.
10
- Using \(t = 0\), explain how the value of \(A\) relates to the experiment.
10
- Show that
$$\log _ { 10 } \theta = \log _ { 10 } A - k t$$
10
- Using Zena's results, calculate the values of \(A\) and \(k\).
10 - Zena states that the temperature of the water will be less than \(1 ^ { \circ } \mathrm { C }\) after 45 minutes. Determine whether the model supports this statement.
10 - Explain why Zena's model is unlikely to accurately give the value of \(\theta\) after 45 minutes.