2013 MI P1 Q9

When a cake is removed from the oven, its temperature decreases at a rate proportional to the positive difference between its temperature and the temperature of the room. The temperature of the room is constant at \(25{}^\circ C\) and \(T\) is the temperature of the cake \(t\) hours after removing from the oven. The temperature of the cake at the instant when it is removed from the oven is \(50{}^\circ C\).

1. Show that \(T=25+25{{e}^{-kt}}\) , where \(k\) is a positive constant.[5]
2. Sketch a solution curve for \(T\) where \(t\ge 0\) .[2]
3. Pradip is a world-renowned baker who is uncompromising on the freshness of the cakes served by his bakery. On a particular morning at 10.00am, he found the temperature of a cake at \(40{}^\circ C\), and \(1\) hour later, it was \(31{}^\circ C\). At what time, to the nearest minute, was the cake removed from the oven?[4]
   State what happens to the temperature of the cake in the long run.[1]