Question

A student must learn m unfamiliar words for an upcoming test. the rate at which the student learns is proportional to the number of items remaining to be learned, with constant of proportionality equal to k. initially, the student knows none of the words. let y(t) stand for the number of the words that the student knows at time t. write down the right hand side of the differential equation satisfied by y. (your answer should be given in terms of y.)

dy/dt =_____

Answer 1

The differential equation that models the student's learning rate is dy/dt = k(m - y), where k is the constant of proportionality, m is the total number of words, and y(t) represents the number of words known at time t.

Step-by-step explanation:

The student's learning rate is proportional to the number of unfamiliar words remaining, which means the rate of learning decreases as the student learns more words.

Since the total number of words to be learned is m, and y(t) represents the number of words known at time t, the number of words remaining to be learned at any time is m - y(t). Therefore, according to the given information, the learning rate can be represented by the differential equation:

dy/dt = k(m - y)

where k is the constant of proportionality.