Course:MATH102/Question Challenge/2008 December Q7

Question

Problem 7: “Live and Learn”

Knowledge can be acquired by studying, but it is forgotten over time A simple model for learning represents the amount of knowledge, y(t), that a person has at time t (in years) by a differential equation

$\displaystyle {\frac {dy}{dt}}=S-fy,$

where $S\geq 0$ is the rate of studying and $f\geq 0$ is the rate of forgetting. We will assume that S and f are constants that are different for each person. [Your answers to the following questions will contain constants such as S or f.]

(a) Mary never forgets anything. What does this imply about the constants S and f? Mary starts studying in school at time t = 0 with no knowledge at all. How much knowledge will she have after 4 years (i.e. at t = 4)?

(b) Tom learned so much in preschool that his knowledge when entering school at time t = 0 is y = 100. However, once Tom is in school, he stops studying completely and only forgets what he knows. What does this imply about the constants S and f?

How long will it take him to forget 75% of what he knew?

(c) Jane studies at the rate of 10 units per year and forgets at rate of 0.2 per year. Sketch a “direction field” (“slope field”) for the differential equation describing Jane’s knowledge. Add a few curves y(t) to show how Jane’s knowledge changes with time.

How much knowledge would Jane have if she keeps studying (and forgetting) for a very long time?

Hints

[show]Hint (a)
(a) If Mary never forgets, which of the constants would be zero for Mary? What would her differential equation then look like?

[show]Hint (b)
(b) If Tom never studies, which of the constants would be zero for him? What would his differential equation then look like?

Solutions

[show]Solution

Part A) Mary never forgets anything. This implies that her $S$ constant is positive and her $f$ constant is equal to zero Therefore: ${\frac {dy}{dt}}=S$ We can find the function $y(t)$ by simple anti differentiation, and we obtain $y(t)=St+y_{0}$. Since Mary starts with no knowledge initially, we have that $y(0)=y_{0}=0$ so that $y(t)=St$. Mary's knowledge after 4 years is given by: $y(4)=S\cdot 4=4S$. Part B) Tom does not study so his $S$ constant is equal to zero. Unlike Mary, Tom is forgetful so his $f$ constant is positive. Therefore: ${\frac {dy}{dt}}=0-fy$ or simply ${\frac {dy}{dt}}=-fy$ Solutions to this differential equation are just exponentially decaying functions of the form: $y(t)=y_{0}e^{-bt}$ We are told that for Tom, $y(t)=y_{0}=100$. Therefore Tom's knowledge is given by: $y(t)=100e^{-ft}$ To find when Tom has forgotten 75% of his knowledge we will isolate for $t$ in the equation below: $25=100e^{-ft}$ $\ln(0.25)=\ln(e^{-ft})$ $-\ln(1/0.25)=-ft$ $ft=\ln(4)$ $t={\frac {\ln(4)}{f}}$ Tom has forgotten 75% of his knowledge after ${\frac {\ln(4)}{f}}$ years. Part C) Jane studies at a rate of 10 units per year and forgets at a rate of 0.2 units per year. Therefore: ${\frac {dy}{dt}}=10-0.2y$ To find the steady states we will set the derivative equal to zero and solve for $y$ $0=10-0.2y$ $0.2y=10$ $y=50$ Slope field for Part (c) A slope field is shown in the diagram. We see that the steady state $y=50$ is stable. Jane will have $y=50$ units of knowledge if she studies and forgets at her current rates for a very long time.