Introduction

Mathematical models are often used to study biological processes, such as population dynamics. In this article, we will analyze a differential equation that describes the rate of change in the population of rabbits. Specifically, we will focus on understanding the population of rabbits on July 1st, using the equation Pt 1000 - 500sin(π/6t), where P is the number of rabbits and t is the beginning of the month, with t0 being January 1st.

Mathematical Model

The given equation is Pt 1000 - 500sin(π/6t). To understand the rate of change in the population of rabbits, we need to find the first derivative of this equation with respect to t.

First Derivative Calculation

The first step in solving this problem is to take the first derivative of the given function with respect to t. Using the chain rule, we differentiate the sine term:

P't -500 times; (π/6) times; cos(π/6t)

This simplifies to:

P't -250π/3 times; cos(π/6t)

Rate of Change on July 1st

July 1st occurs at the beginning of the seventh month, so we substitute t7 into the first derivative equation to determine the rate of change of the rabbit population on this date:

P'7 -250π/3 times; cos(π/6 times; 7)

Simplifying the cosine term:

P'7 -250π/3 times; cos(7π/6)

Since cos(7π/6) -√3/2, the expression becomes:

P'7 -250π/3 times; (-√3/2) 250π√3/6 125π/3

This can be further simplified to approximately 131.3 rabbits per month (three significant figures).

Units of Time

The units of t depend on the context. Here are the different interpretations:

If t is in months: July 1st is at month 7, and the rate of change is 125π/3 rabbits per month. If t is in days: In a non-leap year, July 1st is the 182nd day, and t181. If January 1st is day 0, then the rate of change is 125π/3 rabbits per day. If January 1st is day 1, then the rate of change is also 125π/3 rabbits per day. If t is in seconds: The time at which the rate is measured must be specified. If January 1st 00:00 is 0, then July 1st 00:00 is the specified time, and the rate of change is 125π/3 rabbits per second. If July 1st 00:00 is actual day 182, then the rate is measured for that specific second.

Conclusion

Understanding the rate of change in rabbit populations through mathematical modeling can provide valuable insights into their ecological dynamics. By carefully considering the units of time and the context in which the equation is applied, we can accurately determine the growth or decline rates at specific dates and times.