Question

In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose M denotes the total amount of a subject to be memorized and A(t) is the amount memorized in time t > 0. Determine a differential equation for the amount A(t). (Assume the constant of proportionality is k > 0. Use A for A(t).)dA/dt =

Answer 1

dA/dt=k(M-A)

Explanation:

We need to transform the sentence "the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized" into a differential equation.

So, the rate at which a subject is memorized is equivalent to dA/dt.

Additionally, the amount that is left to be memorized is the total amount of a subject to be memorized less the the amount memorized, so it is equal to:

M - A

Finally, dA/dt and (M-A) are proportionals, so there is a constant k for which it is true that:

dA/dt = k(M-A)

So, A differential equation for the amount A(t) is:

dA/dt = k(M-A)