Dr. MacMillan scoffs at Styrofoam. She is the proud owner of a Sirius Cybernetic
ID: 3004691 • Letter: D
Question
Dr. MacMillan scoffs at Styrofoam. She is the proud owner of a Sirius Cybernetics Corporation thermos (only 35% asbestos!) For this thermos the constant is k = -0.03. Which docs a better job of keeping the coffee warm, the Styrofoam cup or the thermos? How does knowing the value of k allow you to figure out the answer? The next day. Dr. MacMillan leaves Milliway's with a thermos full of coffee at 160 degrees Fahrenheit. It n 8:30 A.M. and the outside temperature is 42 degrees. What is the solution for this particular initial value problem? (Let the time t be measured in minutes, and let t a = 0 stand for 8:30 A.M.) How long must she wait before she is able to drink the coffee? At what time will the coffee fall below 105 degrees and become undrinkable? How much lime docs Dr. MacMillan have to drink her coffee?Explanation / Answer
We can safely assume that coffee follows Newton's law of cooling and the temperature as a function of time is
Tc(t) = Tout + D0e-kt where Tout is outside temperature, Do is initial temperature of coffee and t is time measured in minutes.
4) Note that k has to be positive as coffee cools down with time. Assume k=.03 and if the k of styrofoam > thermos, then coffee does a better job because, it cools down at a slower rate.
For eg, ks = .05, kt = .03, all else being equal, we see that e-.03t > e-.05t
As can bee seen from above, knowing k can help compare the two.
5) Using the equation in 4 and setting t =0, we get
Tc(t) = Tout + D0meaning the temperature of coffee is the initial temperature + the outside temperature.
Substituting Tout = 42 and D = 160, we get Tc(0) = 160 + 42 = 202o F
6) In the absence of data, assume Dr MacMillan burns her tongue if the temperature is above T. So, we have to
solve for how long she has to wait until she can start drinking her coffee.
T = 42 + 160e-.03t
For eg, if T =140, then t= 16.34 minutes.
7) Coffee becomes undrinkable below 105F.
105 = 42 + 160e-.03t. Solving, we get t = 31.067 minutes.
Since beginning time is 8:30 AM, becomes undrinkable at 9:01 AM
8) This is just the difference between the time she can start drinking coffee and before it becomes bad.
In other words, using our example 31.067 - 16.34 = 14.73 minutes
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.