Problem 32 - Calculus and its applications

Question

The undergraduate population at Harbor College was 17,000 and increasing at the rate of \(4.2 \%\) per year. The undergraduate population after \(t\) years, \(P(t)\), is given by \(P(t)=17,000(1.042)^{t}\) Find \(\frac{P(6)-P(2)}{6-2}\). What rate of change does this represent?

Step-by-Step Solution

Answer

The rate of change is the average increase in the population per year from year 2 to year 6.

Step 1: Calculate P(6)

To find the undergraduate population after 6 years, use the given formula. Substitute 6 into the equation: \( P(6) = 17,000 \times (1.042)^{6} \). Calculate \( (1.042)^{6} \) and then multiply the result by 17,000.

Step 2: Calculate P(2)

Similarly, find the undergraduate population after 2 years using the formula. Substitute 2 into the equation:\( P(2) = 17,000 \times (1.042)^{2} \). Calculate \( (1.042)^{2} \) and multiply the result by 17,000.

Step 3: Calculate the Rate of Change

The rate of change is given by the formula \( \frac{P(6) - P(2)}{6 - 2} \). First, subtract \( P(2) \) from \( P(6) \), and then divide the result by 4, which is the difference in years.

Step 4: Interpretation

The result from Step 3 represents the average rate of change of the undergraduate population over the period from year 2 to year 6. This rate of change indicates how much the population increases on average per year over these 4 years.

Key Concepts

Average Rate of Change

Understanding the average rate of change helps students comprehend how quantities vary over a given time period. In this scenario, we are looking at the shift in the undergraduate population at Harbor College over four years. The average rate of change measures the change in population size, dividing it by the number of years to see how the population increased on an average basis. Think of it like checking your speed during a road trip; instead of knowing exactly how fast you're driving every second, you're seeing how far you traveled in a chunk of time.

To calculate the average rate of change, we follow the formula: \[ \frac{P(t_2) - P(t_1)}{t_2 - t_1} \] For this problem, you have \( P(6) \) and \( P(2) \), representing the undergraduate population at years 6 and 2, respectively. By subtracting these two population figures and dividing by the time span of 4 years, you get a meaningful rate that shows the average yearly increase in student population.

This measure is powerful because it offers insight into the trend of population growth over time, even without knowing the exact yearly figures.

Population Growth

Population growth reflects how the number of people in a specific group, like a college, increases over time. It's often expressed as a percentage, indicating how a population size shifts annually under certain conditions. In our Harbor College example, the undergraduate population grows by \(4.2\%\) every year, illustrating an exponential growth process.

• Exponential Growth: This means that the size of the population grows faster as time goes on, creating a curve on the graph instead of a straight line. Each year's increase becomes more substantial because it compounds over time.
• Growth Rate: The growth rate here is \(4.2\%\), serving as a multiplier that the inital population grows by each year. This small number makes a big impact over several years due to the nature of exponential growth.

Bringing this back to the context of Harbor College, the formula \( P(t) = 17,000(1.042)^t \) describes this growth. The initial figure \(17,000\) acts as a starting point, and \( \(1.042\)\) represents the \(4.2\%\) growth rate, a common mathematical trick to model annual percentage growth exponentially.

Mathematical Modeling

Mathematical modeling translates real-world scenarios into mathematical equations. It simplifies complex processes into understandable formulas, allowing easy manipulation and prediction. At its core, it's about using mathematical structures to replicate and anticipate behaviors seen in the world.

For Harbor College's student population, the exponential model \( P(t) = 17,000 \times (1.042)^t \) is used. Let's break this down:

• Initial Population: The number \(17,000\) denotes the starting population.
• Growth Factor: The term \((1.042)^t\) represents the multiplication effect of growth on the original population over time \(t\).

Models like this help administrators plan for future resources, like how many dormitories or teachers they might need. They offer predictive power and insight into how systems behave over time, displaying the tangible benefits of using mathematics to solve real-life challenges.