Question 20 of 20intermediate🔧 ApplyNumerical3 marks

A body cools from 80°C to 50°C in 5 minutes. Calculate the time it takes to cool from 60°C to 30°C. The temperature of the surroundings is 20°C.

Correct Answer

The time taken to cool from 60°C to 30°C is 9 minutes.

Exercise: EXERCISES | Q: 10.20 | (Page 24)
For More Understanding

Explanation

This problem applies Newton's Law of Cooling, which states that for small temperature differences, the rate of cooling is proportional to the temperature difference between the body and its surroundings. The textbook demonstrates this method in Example 10.8, where the average temperature is used to find the temperature difference from surroundings, and the ratio method is applied to find unknown time.

Solution Steps

  1. Step 1: For the first case, calculate the average temperature: (80°C + 50°C)/2 = 65°C

  2. Step 2: Calculate the temperature difference from surroundings: 65°C - 20°C = 45°C

  3. Step 3: Temperature change in first case = 80°C - 50°C = 30°C in 5 minutes

  4. Step 4: For the second case, calculate the average temperature: (60°C + 30°C)/2 = 45°C

  5. Step 5: Calculate the temperature difference from surroundings: 45°C - 20°C = 25°C

  6. Step 6: Temperature change in second case = 60°C - 30°C = 30°C in time t

  7. Step 7: Apply Newton's Law of Cooling: Rate of cooling ∝ temperature difference from surroundings

  8. Step 8: Using the ratio method: (30°C/5 min)/(30°C/t) = 45°C/25°C

  9. Step 9: Simplifying: t/5 = 45/25 = 9/5

  10. Step 10: Therefore, t = 9 minutes

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