A cup of coffee has cooled from 98°C to 60°C after 10 minutes in a room at 23°C. How long will it take to cool to 30°​C?

Question

A cup of coffee has cooled from 98°C to 60°C after 10 minutes in a room at 23°C. How long will it take to cool to 30°​C?

It will take approximately __ minutes.​(Round to three decimal places as​ needed.)

Answer 1

Newton's Law of Cooling states T = T_a+(T₀-T_a)e^-kt where T_a=23°C is the air temperature and T₀=98°C is the initial temperature of the object. Given that the cup of coffee cooled to T=60°C after t=10 minutes, then we can solve for k and use that value to answer how long it'll take it to cool down to 30°C:

T = T_a+(T₀-T_a)e^-kt

60 = 23+(98-23)e^-k(10)

43 = 75e^-10kp

43/75 = e^-10k

ln(43/75) = -10k

(-1/10)ln(43/75) = k

k ≈ 0.0556

Now that we have k, we can then solve for t:

T = T_a+(T₀-T_a)e^-kt

30 = 23+(98-23)e^{-(-1/10)ln(43/75)ˆt}

7 = 75e^{(1/10)ln(43/75)t}

7/75 = e^{(1/10)ln(43/75)t}

ln(7/75) = (1/10)ln(43/75)t

ln(7/75)/[(1/10)ln(43/75)] = t

t ≈ 42.632

Thus, it will take approximately 42.632 minutes for the cup of coffee to cool from 98°C to 30°C.

Hope this helped!

Answer 2

It is not clear if this is a calculus problem for which you need to determine the formula for the temperature of the coffee from a differential equation, or an algebra problem where the formula is given. The first case will require a couple of extra steps, but these steps, stated below, will be pretty meaningless if this is just an algebra problem. I am just going to outline the steps, including the two calculus steps, which you can ignore if not needed. If you would like to see more details to any of the following steps, then let me know what details you would like to see. Note also, that these are the steps required to find the solution. I am not just going to send an answer.

1. (Only needed for calculus students.) Solve the equation referred to as Newton's Law of Cooling for the temperature of the object as a function of time T_ob(t). That is, dT_ob(t)/dt = k ( T_s - T_ob ), where T_s is the temperature of the surroundings and k is a constant of proportionality.
2. (Only needed for calculus students.) Solving the equation in step 1 will generate a constant of integration C, but we are given that T_ob(0) = 98 (assuming t is in minutes and T_ob(t) is in degrees). This condition will allow you to find the value of C. However, it is generally true that C = | T_ob(0) - T_s |
3. Once you have the formula from steps 1 and 2, unless already given, you can use the second piece of information to find the value of k. That is, use the fact that T_ob(10) = 60 to either find the value of k, or to substitute an equivalent expression into the formula for T_ob(t).
4. With k known, you can now substitute T_ob(t) = 30 and solve for t. Note that if we use T_ob(0) = 98, the answer will give the total time it took to cool to 30 from 98. If we just want the time to cool from 60 to 30, either substitute T_ob(0) = 60 in this step, or just subtract 10 minutes.