The temperature of a room is 10°C. A heated object needs 20...

The temperature of a room is 10°C. A heated object needs 20 minutes to reduce its temperature from 80°C to 50°C. Assuming that the temperature of the room is constant and the rate of the cooling of the body is proportional to the difference between the temperature of the heated object and the room temperature. (a) Evaluate the time taken for the heated object to cool down to 30°C. (b) Find the temperature of the object after 50 minutes.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Transcript

00:01 Hello everyone, this question the room temperature t0 is given, t0 is given 10 degrees celsius and they have given the rate of cooling dt by dt equal to minus k t minus t0.

00:25 So we can write this as dt divided by t minus t0 equal to minus k dt.

00:31 So the question here is to find the time t.

00:34 So on integrating both side that is dt t minus t0 minus k into integral of dt, we will be having log of t minus t0 minus kt plus c.

00:48 So this can be written as t minus t0 e to the power minus kt plus c.

00:55 So this can be further written as t minus t0 e to the power minus kt e to the power c.

01:02 Further this can be written as t minus t0 a e to the power minus kt.

01:08 So this implies t equal to t0 plus a t0 plus a e to the power minus kt.

01:16 So mark this as first equation.

01:19 So they have given that at t equal to zero, the temperature is 80 degrees celsius.

01:27 So we will be having 80 t equal to 10 plus a e to the power zero.

01:33 Therefore, 80 plus 10 plus a.

01:36 Therefore, a is 70.

01:38 We have got the value of a.

01:41 Next we find at t equal to 20, the temperature that is t equal to 50 degrees celsius.

01:50 Therefore, 50, 10 plus 70 e to the power minus 20 k.

01:58 Therefore, e to the power minus 20 k will be 40 divided by 70...

Question

Please give Ace some feedback