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 $50{}^\circ C$.
(i)
Show that $T=25+25{{e}^{-kt}}$ , where $k$ is a positive constant.
[5]
(ii)
Sketch a solution curve for $T$ where $t\ge 0$ .
[2]
(iii)
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]
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