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The temperature, \(T\), in degrees Celsius (\(^\circ\)C), of a cup of coffee \(t\) minutes after it is poured is modelled by the function:$$ T(t) = 22 + 70e^{-kt} $$where \(k\) is a positive constant.
  1. Find the initial temperature of the coffee.
  2. The temperature of the coffee is \(65^\circ\)C after 5 minutes. Find the value of \(k\).
  3. Find the temperature of the coffee after 15 minutes.
  4. Find the rate at which the temperature of the coffee is decreasing at \(t=10\) minutes.
  5. State the temperature of the room, giving a reason for your answer.

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