Related questions with answers

As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, a. (a) Let y = f(t) be the fraction of the original material remembered t weeks after the course has ended. Set up a differential equation for y. Your equation will contain two constants; the constant a is less than y for all t. (b) Solve the differential equation. (c) Describe the practical meaning (in terms of the amount remembered) of the constants in the solution y = f(t).

As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, a. Let y=f(t) be the fraction of the original material remembered t weeks after the course has ended. Set up a differential equation for y. Your equation will contain two constants; the constant a is less than y for all t.

At 1:00 pm one winter afternoon, there is a power failure at your house in Wisconsin, and your heat does not work without electricity. When the power goes out, it is $68^{\circ} \mathrm{F}$ in your house. At 10:00 pm, it is $57^{\circ} \mathrm{F}$ in the house, and you notice that it is $10^{\circ} \mathrm{F}$ outside. (a) Assuming that the temperature, T, in your home obeys Newton's Law of Cooling, write the differential equation satisfied by T. (b) Solve the differential equation to estimate the temperature in the house when you get up at 7:00 am the next morning. Should you worry about your water pipes freezing? (c) What assumption did you make in part (a) about the temperature outside? Given this (probably incorrect) assumption, would you revise your estimate up or down? Why?

Question

As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant, a. Describe the practical meaning (in terms of the amount remembered) of the constants in the solution y=f(t).

Solution

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As $t\rightarrow \infty$ then $e^{-kt}\rightarrow 0$ , therefore $y\rightarrow a$ .

Here $a$ represents the fraction of material that is remembered in the long run. The constant $k$ tells us about the rate at which material is forgotten.