Suppose two students are memorizing the elements on a list according to the rate

Question

Suppose two students are memorizing the elements on a list according to the rate of change equation dL/dt = 0.5(1 - L), where L represents the fraction of the list that is memorized at any time t. (a) If one of the students knows one-third of the list at time t = 0 and the other student knows none of the list, which student is learning most rapidly at this instant? Why? (b) What does the rate of change equation predict for someone who begins with the list completely memorized? Explain. (c) Suppose now that the list is infinitely long, like the decimal representation for pi. In reality no one can memorize all the digits to pi, but what does the rate of change equation predict will happen for a person who starts out not knowing any of the digits? That is, according to the rate of change equation, if L = 0 at time t = 0, is there ever a value of t for which L = 1? Explain.

Explanation / Answer

HEre dL/dt = 0.5 (1-L)

(a) if at t= 0 , one student memories 1/3rd of the list so his rate of rememberig the list is

(dL/dT)t =0, L = 1/3 = 0.5 * (1 - 1/3) = 1/3

for students who memorizes none of the list so his rate of remembering the list is

(dL/dT)t =0, L = 0 = 0.5 * (1 - 0) = 1/2

the second student will learn most rapidly and it is because he has nothing else feeded in her mind.

(b) FOr the person who start with list completely memorized where L = 1

(dL/dT)t =0, L =1 = 0.5 * (1 - 1)= 0

so there is zero rate of change as there is nothing else to memorized so learning rate is zero.

(c) By solving the given learning equation

dL/dt = 0.5 (1-L)

we get,

dL/ (1-L) = 0.5 dt

[-ln (1-L)]0L = 0.5 t

L = 1 - e-0.5t

so at t = infiinity we will be able to earn the complete list.

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