Chapter 9.3, Problem 80E

In 1960, William K. Estes proposed the following model for measuring a student’s performance in the classroom:

$\frac{dk}{dt}+\mathcal{l}k=\lambda \mathcal{l}$

where k(t) is the student’s knowledge after t weeks (expressed as a percentage and measured by performance on examinations), ℓ is a constant called the coefficient of learning and representing the student’s ability to learn (expressed as a percentage and determined by IQ or some similar general intelligence predictor), and λ is a constant representing the fraction of available time the student spends performing helpful acts that should increase knowledge of the subject (studying, going to class; and so on). Use this model to solve Problems 79 and 80.

80.    Learning theory. Refer to Problem 79. When will both students have the same level of knowledge? Round answer to one decimal place.

79.    Learning theory. Students enrolled in a beginning Spanish class are given a pretest the first day of class to determine their knowledge of the subject. The results of the pretest, the coefficient of learning, and the fraction of time spent performing helpful acts for two students in the class are given in the table. Use the Estes model to predict the knowledge of each student after 6 weeks in the class.