Problem 164 - Calculus Volume 1

Question

A small town in Ohio commissioned an actuarial firm to conduct a study that modeled the rate of change of the town’s population. The study found that the town’s population (measured in thousands of people) can be modeled by the function \(P(t)=-\frac{1}{3} t^{3}+64 t+3000,\) where \(t\) is measured in years. a. Find the rate of change function P?(t) of the population function. b. Find P?(1), P?(2), P?(3), and P?(4). Interpret what the results mean for the town. c. Find P??(1), P??(2), P??(3), and P??(4). Interpret what the results mean for the town’s population.

Step-by-Step Solution

Answer

The rate of change is decreasing, indicating population growth slowdown.

Step 1: Find the rate of change function

To find the rate of change of the population function, we need to calculate the first derivative \(P'(t)\) of \(P(t) = -\frac{1}{3} t^{3} + 64t + 3000\).Differentiate each term:- The derivative of \(-\frac{1}{3}t^3\) is \(-t^2\).- The derivative of \(64t\) is \(64\).- The derivative of the constant \(3000\) is \(0\).Thus, the rate of change function is \(P'(t) = -t^2 + 64\).

Step 2: Calculate the rate of change at specific years

Now, evaluate \(P'(t)\) at \(t = 1, 2, 3,\) and \(4\):- \(P'(1) = -(1)^2 + 64 = -1 + 64 = 63\)- \(P'(2) = -(2)^2 + 64 = -4 + 64 = 60\)- \(P'(3) = -(3)^2 + 64 = -9 + 64 = 55\)- \(P'(4) = -(4)^2 + 64 = -16 + 64 = 48\)This means that the population is initially increasing by 63,000 people per year and the growth rate decreases each year.

Step 3: Find the concavity of the population function

To understand how the rate of change itself is changing, we need the second derivative \(P''(t)\) of \(P(t)\). Differentiate \(P'(t) = -t^2 + 64\):- The derivative of \(-t^2\) is \(-2t\).- The derivative of \(64\) is \(0\).Thus, \(P''(t) = -2t\).

Step 4: Calculate the concavity at specific years

Evaluate \(P''(t)\) at \(t = 1, 2, 3,\) and \(4\):- \(P''(1) = -2 \times 1 = -2\)- \(P''(2) = -2 \times 2 = -4\)- \(P''(3) = -2 \times 3 = -6\)- \(P''(4) = -2 \times 4 = -8\)The negative values show that the rate of population growth is decreasing over time, indicating a slow down in growth each year.

Key Concepts

Derivative

A derivative refers to the rate at which a function is changing at any given point. In simple terms, it's a way to measure how a quantity is changing over time.
For our study, the town's population is described by the function:

• \( P(t) = -\frac{1}{3} t^3 + 64t + 3000 \)

To find how the population changes, we calculate the first derivative, \( P'(t) \), of this function.
The derivative helps us understand the rate at which the population is growing or shrinking. It's the foundation for determining trends over time.

Rate of Change

The rate of change measures how quickly a quantity is increasing or decreasing. In our case, it's the change in population from year to year. We used the first derivative, \( P'(t) \), to find the rate of change.

• \( P'(t) = -t^2 + 64 \)

When we plug in specific years like 1, 2, 3, and 4, we learn how the population growth is adjusting each year.
For example:

• At \( t = 1 \), the population increases by 63,000 people/year.
• At \( t = 4 \), the growth slows to 48,000 people/year.

This shows that although the population is growing, the speed of growth declines over these years.

Second Derivative

The second derivative, \( P''(t) \), tells us how the rate of change itself is changing. It's the derivative of the first derivative and gives insights into the acceleration or deceleration of population growth.
In our model:

• \( P''(t) = -2t \)

By calculating \( P''(t) \) at different times, we understand how the growth rate is slowing down.

• At \( t = 1 \), \( P''(t) = -2 \), and this negative value indicates deceleration.
• At \( t = 4 \), \( P''(t) = -8 \), showing stronger deceleration.

These results reveal that while the population keeps growing, the rate at which it grows is decreasing.

Concavity

Concavity refers to whether a function is curving upwards or downwards. Using the second derivative helps us determine this property.
If \( P''(t) \) is:

• Positive, the function is concave up (like a cup).
• Negative, the function is concave down (like a frown).

In the case of our town's population model:

• \( P''(t) = -2t \) is always negative for \( t > 0 \).
• This means the population function is always concave down for positive time values.

The downward curvature suggests a consistent slowing of population growth.

Population Modeling

Population modeling involves creating mathematical functions to predict how a population might change over time. These models can help towns like ours plan for the future.

• **Initial Function**: \( P(t) = -\frac{1}{3}t^3 + 64t + 3000 \)
• This model predicts both the population size and its rate of growth.
• **Analyzing Derivatives**: By studying the first and second derivatives, we understand not just the growth, but how the growth's speed evolves.

Population models are crucial for urban planning, helping communities decide on resource allocation, infrastructure needs, and more.
Through calculus, these models provide insights into future trends, making it a vital tool for decision-makers.