Question

If water at \(100^{\circ}\text{C}\) cools in 10 minutes to \(80^{\circ}\text{C}\) and to \(65^{\circ}\text{C}\) in the next 10 minutes, then the room temperature will be ...

• A. \(30^{\circ}\text{C}\)
• B. \(15^{\circ}\text{C}\)
• C. \(25^{\circ}\text{C}\)
• D. \(20^{\circ}\text{C}\)

Hint:

Newton law gives a geometric progression of temperature differences over equal times.

Answer 1

The Correct Option is D

Step 1: Exponential decay
The excess temperature follows $\theta = \theta_0e^{-kt}$. In equal intervals it is multiplied by the same factor $r = e^{-10k}$.

Step 2: Use the data
$80-s = r(100-s)$ and $65-s = r(80-s)$. Subtracting the second from the first gives $15 = r(20)$, so $r = \frac34$. Then $80-s = \frac34(100-s)$ gives $320-4s = 300-3s$, so $s = 20$.

Step 3: Direct check
Test $s=20$: excess $80, 60, 45$, common ratio $0.75$. Test other options, such as $s=15$, giving $85, 65, 50$ with ratios $0.765$ and $0.769$, not equal. So $s=20$, option (D).

Final Answer:
20 degrees Celsius. \[ \boxed{\text{(D)}\ 20^{\circ}\text{C}} \]