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)

And the solution for this equation is:

Using the substitution u = M-A, we have that du = -dA and after integrate we got:

C is a constant.

Now we can apply exponential in both sides and we got:

Step-by-step explanation:

Let's define some notation first :

represent the total amount of the subject to be memorized

represent the amount memorized at time t

t represent the time

k is a constant of proportionality

For this case we know that the rate of memorizing is proportional to the amount left to be memorized.

So then we can create the following relationship:

Where k>0. We can solve this differential equation like this:

Using the substitution u = M-A, we have that du = -dA and after integrate we got:

C is a constant.

Now we can apply exponential in both sides and we got:

answer: -1

step-by-step explanation: