11 Answer part (iii) of this question on the insert provided.
A hot drink is made and left to cool. The table shows its temperature at ten-minute intervals after it is made.
| Time (minutes) | 10 | 20 | 30 | 40 | 50 |
| Temperature \(\left( { } ^ { \circ } \mathrm { C } \right)\) | 68 | 53 | 42 | 36 | 31 |
The room temperature is \(22 ^ { \circ } \mathrm { C }\). The difference between the temperature of the drink and room temperature at time \(t\) minutes is \(z ^ { \circ } \mathrm { C }\). The relationship between \(z\) and \(t\) is modelled by
$$z = z _ { 0 } 10 ^ { - k t }$$
where \(z _ { 0 }\) and \(k\) are positive constants.
- Give a physical interpretation for the constant \(z _ { 0 }\).
- Show that \(\log _ { 10 } z = - k t + \log _ { 10 } z _ { 0 }\).
- On the insert, complete the table and draw the graph of \(\log _ { 10 } z\) against \(t\).
Use your graph to estimate the values of \(k\) and \(z _ { 0 }\).
Hence estimate the temperature of the drink 70 minutes after it is made.