A cup of coffee is cooling down in a room. At time \(t\) minutes after the coffee is made, its temperature is \(x ^ { \circ } \mathrm { C }\), where
$$x = 15 + 70 \mathrm { e } ^ { - \frac { t } { 40 } }$$
Find the temperature of the coffee when it is made.
Find the temperature of the coffee 30 minutes after it is made.
Find how long it will take for the coffee to cool down to \(60 ^ { \circ } \mathrm { C }\).
Use integration to solve the differential equation
$$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \frac { 1 } { 40 } ( x - 15 ) , \quad x > 15$$
given that \(x = 85\) when \(t = 0\), expressing \(t\) in terms of \(x\).
Hence show that \(x = 15 + 70 \mathrm { e } ^ { - \frac { t } { 40 } }\).