Question

The initial temperature of a body is \(80^{\circ }\text{C}\). If its temperature falls to \(64^{\circ }\text{C}\) in 5 minutes and in the next 10 minutes to \(52^{\circ }\text{C}\), then the temperature of the surroundings is

• A. \(35^{\circ }\text{C}\)
• B. \(42.6^{\circ }\text{C}\)
• C. \(48^{\circ }\text{C}\)
• D. \(26^{\circ }\text{C}\)

Hint:

By Newton's law of cooling, the excess temperature over the surroundings is multiplied by the same factor in equal times.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The excess temperature \(\theta-\theta_0\) decays by the same factor \(k\) in every 5 minutes.

Step 2: Equations
First 5 min: \(64-\theta_0=k(80-\theta_0)\).
Next 10 min (two steps): \(52-\theta_0=k^2(64-\theta_0)\).

Step 3: Eliminate k
Let \(x=64-\theta_0\). Then \(k=\dfrac{x}{x+16}\) and \(k^2=\dfrac{x-12}{x}\). This gives \(5x^2-32x-768=0\), so \(x=16\) (the other root is negative).

Step 4: Result
\(\theta_0=64-16=48^{\circ}\text{C}\).

Step 5: Check
The excess goes 32, 16, 4: halved in 5 min, then halved twice in the next 10 min, which is consistent.

Final Answer:
The temperature of the surroundings is 48 degrees Celsius. \[ \boxed{\text{(C) }48^{\circ}\text{C}} \]