Problem 68 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

PopULATION GRoWTH A study prepared for a Sunbelt town's chamber of commerce projected that the town's population in the next 3 yr will grow according to the rule $$ P(t)=50,000+30 t^{3 / 2}+20 t $$ where \(P(t)\) denotes the population \(t\) mo from now. How fast will the population be increasing \(9 \mathrm{mo}\) and \(16 \mathrm{mo}\) from now?

Step-by-Step Solution

Answer

The population of the town will be increasing at a rate of 155 people per month after 9 months and 200 people per month after 16 months.

Step 1: Differentiate the population growth function with respect to time.

Recall that the population function is given by: $$ P(t) = 50,000 + 30t^{3/2} + 20t $$ Differentiating with respect to time, we get: $$ \frac{dP}{dt} = \frac{d}{dt}(50,000 + 30t^{3/2} + 20t) $$ Using the power rule and the constant rule, we get: $$ \frac{dP}{dt} = 0 + 30\cdot\frac{3}{2}t^{1/2} + 20 $$ Simplifying, we find the rate of population change as: $$ \frac{dP}{dt} = 45t^{1/2} + 20 $$

Step 2: Evaluate the rate of population change after 9 months.

We are asked to find how fast the population will be increasing 9 months from now. To do this, evaluate the rate of population change when \(t = 9\): $$ \frac{dP}{dt}(9) = 45(9)^{1/2} + 20 $$ Calculating the numerical value, we get: $$ \frac{dP}{dt}(9) = 45(3) + 20 = 155 $$ So, the population will be increasing at a rate of 155 people per month after 9 months.

Step 3: Evaluate the rate of population change after 16 months.

We are also asked to find how fast the population will be increasing 16 months from now. To do this, evaluate the rate of population change when \(t = 16\): $$ \frac{dP}{dt}(16) = 45(16)^{1/2} + 20 $$ Calculating the numerical value, we get: $$ \frac{dP}{dt}(16) = 45(4) + 20 = 200 $$ So, the population will be increasing at a rate of 200 people per month after 16 months. In summary, the population of the town will be increasing at a rate of 155 people per month after 9 months and 200 people per month after 16 months.

Key Concepts

Differential Calculus

Differential calculus is a branch of mathematics that deals with the study of how things change. It's all about understanding the rates at which quantities vary and it provides us with the tools to calculate these rates. In our exercise context, we differentiated a given population growth function, using rules such as the power rule (which tells us how to differentiate expressions like t^{n}) and the constant rule (which tells us that the derivative of a constant is zero).
In simpler terms, think of differential calculus as a mathematical microscope that allows us to see and measure how fast things are changing at any given moment. For the population of the Sunbelt town, we used it to calculate how quickly the number of people in the town is expected to increase.

Rate of Change

The rate of change is a key concept in calculus and is especially important in understanding real-world problems like population growth. It refers to how much a quantity, such as population, increases or decreases over a certain period of time. Calculating the rate of change gives us a snapshot of the speed at which conditions are evolving at a specific moment.
In our exercise, the rate of population change is represented by \( \frac{dP}{dt} \), which is the derivative of the population function with respect to time. Intuitively, it's like looking at the speedometer of a car to see how fast it's going at an exact moment, instead of just knowing where it started and where it ended up. By finding the value of \( \frac{dP}{dt} \) at specific times, we can tell exactly how fast the population of the town is growing at those times.

Population Modeling

Population modeling is a mathematical method used to represent the dynamics of the population of organisms or people within a certain area. These models are crucial for planning, understanding environmental impacts, and making economic decisions. Our exercise used a specific equation to model the population growth of a town.
The equation P(t)=50,000+30t^{3/2}+20t is a model that estimates the population size based on time. While it may look complex, at its heart, it simply predicts that the population will increase from its current size (50,000) based on two contributions - one that grows quickly (as t increases to the power of 3/2) and a linear term that represents steady growth over time (proportional to t). By differentiating this model, we can discuss how the population growth rate changes over time, which in turn helps the town’s chamber of commerce to make informed decisions about resources and infrastructure.