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

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

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

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.