Question

4. A student studying a foreign language has 50 verbs to memorize. The rate at which the
student can memorize these verbs is proportional to the number of verbs remaining to be
memorized, 50 – y, where the student has memorized y verbs. Assume that initially no verbs
have been memorized and suppose that 20 verbs are memorized in the first 30 minutes.
(a) How many verbs will the student memorize in two hours?
(b) After how many hours will the student have only one verb left to memorize?

Answer 1

The differential equation here is, where theis a proportionality constant. This equation is separable, so we may solve by integration:

Now we may solve for the constants. When (no verbs have beenmemorized). So we have

So we have now .

Now solve for: (We'll assume that the time is in hours so that for the first 20verbs.)

. Solvingthis (by using natural logarithms) gives

Sothe solution is

Now we may "plug in" for part a:

For part b, set (1 word left is 49 words memorized).