Question

On a hot day, students are lining up to buy ice cream. Let L be the number of people in line. Write a differential equation for L using the following assumptions:

Students join the line at a rate proportional to the number of people already in line, with a proportionality constant of 0.1.
Students get ice cream and leave the line at a constant rate of 0.4 people per minute.
Students get tired of standing in line and leave at a per capita rate proportional to the number of people in line, with a proportionality constant of 0.02.
Write a differential equation for L.

L′=_______

Answer 1

The differential equation for the number of students in line to buy ice cream, with the rate of joining and leaving the line accounted for, is L' = 0.08L - 0.4.

Step-by-step explanation:

To write a differential equation for the number of people in line to buy ice cream, denoted as L, we consider the given assumptions:

• Students join the line at a rate proportional to the number of people already in line (proportionality constant of 0.1).
• Students get ice cream and leave the line at a constant rate (0.4 people per minute).
• Students get tired of standing in line and leave at a rate proportional to the number of people in line (proportionality constant of 0.02).

The differential equation for L can be expressed as:

L' = 0.1L - 0.4 - 0.02L

Consolidating the terms involving L gives:

L' = (0.1 - 0.02)L - 0.4

L' = 0.08L - 0.4

This equation represents the rate of change of the number of students in line with respect to time.