- A teacher made a cup of coffee. The temperature \(\theta ^ { \circ } \mathrm { C }\) of the coffee, \(t\) minutes after it was made, is modelled by the differential equation
$$\frac { \mathrm { d } \theta } { \mathrm {~d} t } + 0.05 ( \theta - 20 ) = 0$$
Given that
- the initial temperature of the coffee was \(95 ^ { \circ } \mathrm { C }\)
- the coffee can only be safely drunk when its temperature is below \(70 ^ { \circ } \mathrm { C }\)
- the teacher made the cup of coffee at 1.15 pm
- the teacher needs to be able to start drinking the coffee by 1.20 pm
use two iterations of the approximation formula
$$\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }$$
to estimate whether the teacher will be able to start drinking the coffee at 1.20 pm .