Solve only the blank that is left open!!!!! If right you will recieve 5 stars!!

Question

Solve only the blank that is left open!!!!! If right you will recieve 5 stars!!

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, using k as any constant of proportionality you may need (let k > 0). Your equation will contain two constants; the constant a (also positive) is less than y for all t. dy/dt What is the initial condition for your equation?y0 Solve the differential equation. y = a + (1-a)e(-kt) What are the practical meaning (in terms of the amount remembered) of the constants in the solution y = f(i)? If after one week the student remembers 75 percent of the material learned in the semester, and after two weeks remembers 61 percent, how much will she or he remember after summer vacation (about 14 weeks)? percents EK You can earn partial credit on this problem.

Explanation / Answer

so we know

0.75 = a + (1-a) e^(-k*1)
so
e^(-k) = (0.75-a)/(1-a)

and
0.61 = a + (1-a) e^(-k*2)

so
0.61 = a + (1-a) * ((0.75-a)/(1-a))^2

a=19/44

then plug in to find k

e^(-k) = (0.75-19/44)/(1-19/44)
k= 0.5798

so
at 14 weeks

19/44 + (1-19/44)*e^(-0.5798*14)
=0.432
or 43.2 %

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