# Temperature Problem

1. May 22, 2006

### cscott

The temperature of a body differs from that of a medium, whose temperature is kept constant, by 40 °C. In 5 min, this difference is 20 °C. (a) What is the value of k? (b) In how many minutes will the difference in temperature be 10 °C?

$$T_B-T_M = \left ( T_0-T_M \right )e^{-kt}$$

Do we know k is negative only because of what's said in part (b)?

2. May 22, 2006

### Hammie

I would say that the value of k is negative because it is the difference in temperature between the object and environment that is decreasing with time. It wouldn't make any difference whether the object was initially hotter or cooler than its surrrounding environment.

3. May 22, 2006

### cscott

Yeah, I should have quoted the part dealing with 5 minutes later. Thanks.

4. May 22, 2006

### HallsofIvy

Staff Emeritus
Actually we know that k is positive (so that -k is negative) because of the fact that the difference between the two temperatures is decreasing (from 40 to 20 in 5 minutes).

After 5 minutes, TB- TM= 20 and T0- TM= 40 so the equation, with t= 5 says
20= 40e-5k. It should be easy to solve that for k and then solve 10= 40e-kt, with that value of k, for t.

By the way, you don't really need to answer (a) in order to answer (b) (although you do have to answer (a) anyway!). If the temperature difference drops from 40 to 20 in 5 minutes, then it halves in in 5 minutes. In another 5 minutes, it will halve again! (That's a basic property of exponentials).