Problem 54 - Calculus

Question

The population of a herd of deer is modeled by $$P(t)=4000+500 \sin \left(2 \pi t-\frac{\pi}{2}\right)$$ where \(t\) is measured in years from January 1 (a) How does this population vary with time? Sketch a graph of \(P(t)\) for one year. (b) Use the graph to decide when in the year the population is a maximum. What is that maximum? Is there a minimum? If so, when? (c) Use the graph to decide when the population is growing fastest. When is it decreasing fastest? (d) Estimate roughly how fast the population is changing on the first of July.

Step-by-Step Solution

Answer

The population varies sinusoidally with a minimum of 3500 and a maximum of 4500. Peak growth occurs mid-year (July 1st), and maximal decline at the start (January 1st). On July 1st, it’s growing at ~3142 deer/year.

Step 1: Understanding the Population Function

The function given is \( P(t) = 4000 + 500 \sin\left(2 \pi t - \frac{\pi}{2}\right) \). This is a sinusoidal function with a mean (or equilibrium position) at 4000, an amplitude of 500, and a frequency of 1 cycle per year, due to the \( 2\pi \) coefficient of \( t \). The phase shift is \( \frac{\pi}{2} \) to the right.

Step 2: Graphing the Function for One Year

For one year, since the function completes one cycle per year, plot \( t \) from 0 to 1. At \( t = 0 \), \( \sin\left(2\pi \times 0 - \frac{\pi}{2}\right) = -1 \), leading to \( P(0) = 4000 - 500 = 3500 \). At \( t = \frac{1}{4} \), \( P\left(\frac{1}{4}\right) = 4000 + 500 = 4500 \). At \( t = \frac{1}{2} \), \( P\left(\frac{1}{2}\right) = 4000 + 0 = 4000 \). At \( t = \frac{3}{4} \), \( P\left(\frac{3}{4}\right) = 4000 - 500 = 3500 \). At \( t = 1 \), \( P(1) = 4000 + 500 = 4500 \). Thus, the graph is a sinusoidal wave starting at 3500, peaking at 4500, and returning to 3500.

Step 3: Maximum and Minimum Population

The maximum value of \( P(t) \) occurs when \( \sin\left(2\pi t - \frac{\pi}{2}\right) = 1 \), which is at \( t = \frac{1}{4} \) and \( t = 1 \) (April 1st and December 31st). The maximum population is 4500. The minimum occurs when \( \sin\left(2\pi t - \frac{\pi}{2}\right) = -1 \), which is at \( t = 0.75 \) (October 1st), and it is 3500.

Step 4: Population Change Rate

The rate of change is given by the derivative, \( P'(t) = 1000\pi \cos\left(2\pi t - \frac{\pi}{2}\right) \). The population grows fastest when \( \cos\left(2\pi t - \frac{\pi}{2}\right) = 1 \), which is at \( t = 0.5 \) (July 1st). It decreases fastest when \( \cos\left(2\pi t - \frac{\pi}{2}\right) = -1 \), at \( t = 0 \) (January 1st).

Step 5: Estimating Population Change on July 1st

On July 1st (\( t = 0.5 \)), the derivative \( P'(0.5) = 1000\pi \times 1 = 1000\pi \). Thus, the population is changing at a rate of approximately 3142 deer per year.

Key Concepts

Population Modeling

Population modeling is essential in understanding how a group of individuals in a specific area, such as a herd of deer, changes over time. In the given exercise, population is modeled by a sinusoidal function: \[ P(t) = 4000 + 500 \sin\left(2\pi t - \frac{\pi}{2}\right) \]This equation helps predict the number of deer at any point in time.

• The number 4000 is the average population, meaning the population hovers around this number.
• The amplitude 500 represents how much the population deviates above and below the average during its cycle.
• The presence of the sine function indicates that the population changes periodically, a common trait in natural processes.

This function can account for seasonal factors affecting animal populations, like food availability or breeding patterns.
Understanding these factors through mathematical modeling is crucial for wildlife management and conservation efforts.

Rate of Change

The rate of change in this context refers to how fast the deer population increases or decreases at any given point in the year. To find this, we take the derivative of the population function, which tells us how the population is changing instantaneously.
For the function: \[ P(t) \], the derivative is: \[ P'(t) = 1000\pi \cos\left(2\pi t - \frac{\pi}{2}\right) \]This derivative tells us:

• The population grows fastest when the derivative reaches its maximum positive value, specifically when \( \cos(2\pi t - \frac{\pi}{2}) = 1 \).
• It decreases fastest when the derivative is at its most negative, \( \cos(2\pi t - \frac{\pi}{2}) = -1 \).
• At zero crossings of the cosine function, the population is neither increasing nor decreasing rapidly.

For practical examples, on July 1st (\( t = 0.5 \)), the population increases rapidly at a rate of \( 1000\pi \approx 3142 \) deer per year.
Understanding rates of change helps in anticipating and managing shifts in population density.

Graphing Periodic Functions

Graphing periodic functions helps visualize how attributes such as amplitude, frequency, and phase shifts affect the behavior of a function over time.
Consider the population function given:\[ P(t) = 4000 + 500 \sin\left(2\pi t - \frac{\pi}{2}\right) \]When graphing:

• The function completes one full cycle over the year, with important periods at the start, quarter, and halfway through the year.
• The cycle begins at a population of 3500, reaches a peak of 4500 by mid-year, goes back to 4000, and returns to 3500 as it completes the cycle.
• Key points on the graph correspond to significant calendar dates:
  • Maximum population on April 1st and December 31st.
  • Minimum population on October 1st.

By visualizing the function, students can better understand how the population trends across months and seasons, paving the way for more intuitive interpretations.
Graphing these functions sheds light not only on patterns but also on anomalies and changes associated with external factors.