In a theory of learning, the rate at which words are memorized by an average per

Question

In a theory of learning, the rate at which words are memorized by an average person
is assumed to be proportional to the square root of the number of words that is left to be memorized. Suppose that there are 100 words to be memorized and y(t) is the amount memorized after time t (in hours). Assume the proportional constant is k = 4.
(i) Determine a dierential equation for y(t) and identify its type;
(ii) Solve the dierential equation;
(iii) What is the initial value y(0) ? How long does it take to memorize all 100 words ?

Explanation / Answer

Words of average person = w

dw/dt = k(100-w)

i) Seperable nonlinear differential eqn of the 1st order

dw/(100-w) = k dt

Integrate both sides

-2 (100-w)1/2 = kt +c

100-w = (-4t/2-c/2)2

w =100- (2t+c/2)2

w(0) = 100 = > c=0

ii) w = 100-(2t)^2

iii)

w(0) = y(0) = 100

0 = 100 -(2t)^2

100 = (2t)^2

10 = 2t

t = 5 hours

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