8 In an experiment, the temperature of a hot liquid is measured every minute.
The difference between the temperature of the hot liquid and room temperature is \(D ^ { \circ } \mathrm { C }\) at time \(t\) minutes.
Fig. 8 shows the experimental data.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-07_1144_1541_497_276}
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\caption{Fig. 8}
\end{figure}
It is thought that the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) might fit the data.
- Write down the derivative of \(\mathrm { e } ^ { - 0.03 t }\).
- Explain how you know that \(70 \mathrm { e } ^ { - 0.03 t }\) is a decreasing function of \(t\).
- Calculate the value of \(70 \mathrm { e } ^ { - 0.03 t }\) when
- \(\quad t = 0\),
- \(t = 20\).
- Using your answers to parts (b) and (c), discuss how well the model \(D = 70 \mathrm { e } ^ { - 0.03 t }\) fits the data.