MathProblem 24 Baby bottle problem

You need to warm milk in a baby bottle from its initial temperature of 15 degrees centigrade to 25 degrees. You put the bottle in a pot of boiling water which stays at constant temperature of 100 degrees. The thickness and conductivity of the bottle are such that the initial rate of heat transfer is 85 degrees per minute. However heat transfer is proportional to the difference between the temperature of the milk and the water. How many minutes will it take to heat the bottle to 25 degrees?

Solution

降温速率与温度差有关，不妨设速率为 \(r\), 设温度为 \(t\):

\[r = (100-t) \]

现考虑升温从 \(t\rightarrow t+\delta t\), 由此可知此时花费的时间：

\[\delta T = \frac{t+\delta t- t}{100-t}=\frac{\delta t}{100-t} \]

对上式进行积分即可：

\[T=\int_{t=15}^{25}\frac{1}{100-t}dt \]