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A bowl filled with very hot soup cools from 98°C to 86°C in 2 minutes when the room temperature is 22°C. How long it will take to cool from 75°C to 69°C?
$$\frac{dT}{dt} = -k(T - T_s)$$
For small temperature changes: $$\frac{T_1 - T_2}{t} = k \left( \frac{T_1 + T_2}{2} - T_s \right)$$
$$\frac{12}{2} = k(92 - 22)$$
$$6 = k(70) \implies k = \frac{6}{70} = \frac{3}{35}$$
$$\frac{6}{t'} = k(72 - 22)$$
$$\frac{6}{t'} = \frac{150}{35} = \frac{30}{7}$$
$$t' = \frac{6 \times 7}{30} = \frac{42}{30} = 1.4 \text{ minutes}$$
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