Newton's Law of Cooling differential equation examples explained

📚 Quick Study Guide

• 🌡️ Newton's Law of Cooling describes the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings).
• 📝 The differential equation representing Newton's Law of Cooling is: $\frac{dT}{dt} = k(T - T_s)$, where:
  • $T$ is the temperature of the object at time $t$.
  • $T_s$ is the surrounding temperature (assumed constant).
  • $k$ is a constant that depends on the properties of the object.
• ➗ The general solution to the differential equation is: $T(t) = T_s + (T_0 - T_s)e^{kt}$, where $T_0$ is the initial temperature of the object.
• ➕ Note that $k$ is usually negative, indicating that the object's temperature decreases over time if $T_0 > T_s$.
• 💡 To solve problems using Newton's Law of Cooling, you often need to determine the value of $k$ using given data points.

Practice Quiz

1. A cup of coffee has an initial temperature of 95°C. The room temperature is 20°C. After 10 minutes, the coffee's temperature is 60°C. What is the temperature after 20 minutes?
   1. 35°C
   2. 45°C
   3. 40°C
   4. 50°C
2. A metal rod is heated to 100°C and then placed in a room at 25°C. After 5 minutes, the rod's temperature is 60°C. Find the value of $k$.
   1. $\frac{1}{5}ln(\frac{75}{35})$
   2. $5ln(\frac{75}{35})$
   3. $\frac{1}{5}ln(\frac{35}{75})$
   4. $5ln(\frac{35}{75})$
3. A pie is taken out of the oven at 180°C and placed in a room at 25°C. After 30 minutes, the pie's temperature is 80°C. What was the temperature of the pie after 10 minutes?
   1. 120°C
   2. 140°C
   3. 131.6°C
   4. 150°C
4. An object cools from 80°C to 60°C in 20 minutes when the surrounding temperature is 25°C. How long will it take to cool to 40°C?
   1. 45.1 minutes
   2. 30 minutes
   3. 50.5 minutes
   4. 60 minutes
5. A thermometer is taken from a room where the temperature is 75°F to the outdoors, where the temperature is 35°F. After 1 minute, the thermometer reads 65°F. What will it read after another minute?
   1. 57°F
   2. 55°F
   3. 58°F
   4. 59°F
6. A bottle of soda at 70°F is placed in a refrigerator where the temperature is 40°F. After half an hour, the soda has cooled to 60°F. What is the temperature of the soda after another half hour?
   1. 54°F
   2. 50°F
   3. 52°F
   4. 55°F
7. The temperature of a hot potato placed in a room at 20°C is observed to be 60°C after 12 minutes. The initial temperature of the potato was 90°C. What is the value of 'k'?
   1. $\frac{1}{12}ln(\frac{7}{4})$
   2. $12ln(\frac{7}{4})$
   3. $\frac{1}{12}ln(\frac{4}{7})$
   4. $12ln(\frac{4}{7})$