Problem
Newton's Law of Cooling (or Heating) is used to estimate the total time needed to heat treat a part for jet engines. The cata below are obtained with $t$ in hours and $T(t)$ in degrees centigrade. \[ T(3)=118 \text { and } T^{\prime}(3)=1.4 \] Which of the following best describes what $T^{\prime}(3)=1.4$ indicates? The temperature of the part is increasing at a rate of 1.4 degrees centigrade per hour. From the third to the fourth hour the temperature of teh part will increase by 1.4 degrees centigrade. The temperature of the part is 118 degrees centigrade after three hours. Newton's Law of Cooling cannot be used since this situation is an increase in temperature. While all of these are delightful answers, none of them are accurate.
Solution
Step 1 ：Given the data from Newton's Law of Cooling (or Heating) with $t$ in hours and $T(t)$ in degrees centigrade: $T(3)=118$ and $T^{\prime}(3)=1.4$.
Step 2 ：We are asked to interpret what $T^{\prime}(3)=1.4$ indicates.
Step 3 ：The derivative $T^{\prime}(3)=1.4$ represents the rate of change of the temperature with respect to time at $t=3$ hours.
Step 4 ：Thus, $T^{\prime}(3)=1.4$ indicates that the temperature of the part is increasing at a rate of 1.4 degrees centigrade per hour.
Step 5 ：\(\boxed{\text{Final Answer: The temperature of the part is increasing at a rate of 1.4 degrees centigrade per hour.}}\)
