Problem 11 - Applied Calculus

Question

A city's population was 30,700 in the year 2000 and is growing by 850 people a year. (a) Give a formula for the city's population, \(P\), as a function of the number of years, \(t\), since 2000 . (b) What is the population predicted to be in 2010 ? (c) When is the population expected to reach 45,000 ?

Step-by-Step Solution

Answer

(a) \( P(t) = 30700 + 850t \). (b) Population in 2010: 39,200. (c) Year 2017.

Step 1: Write the Initial Information

We know that the initial population in the year 2000 is 30,700 and it grows by 850 people each year. We can express the population as a linear function of the number of years since 2000.

Step 2: Formulate the Function

To express the population as a function of years since 2000, we start with the initial population and add the product of the growth rate and the number of years. The formula is given by: \[ P(t) = 30700 + 850t \]where \( P(t) \) is the population after \( t \) years.

Step 3: Predict Population in 2010

To find the population in 2010, calculate the number of years since 2000, which is 10 years. Substitute \( t = 10 \) into the function:\[ P(10) = 30700 + 850 \times 10 \]

Step 4: Compute Population for 2010

Calculate the expression from Step 3:\[ P(10) = 30700 + 8500 = 39200 \]Thus, the population in 2010 is predicted to be 39,200.

Step 5: Determine When Population Reaches 45,000

To find when the population reaches 45,000, set \( P(t) = 45000 \) in the population formula and solve for \( t \):\[ 30700 + 850t = 45000 \]

Step 6: Solve for \( t \)

Rearrange the equation from Step 5 to solve for \( t \):\[ 850t = 45000 - 30700 \]Calculate the right side:\[ 850t = 14300 \]Now, divide both sides by 850:\[ t = \frac{14300}{850} \approx 16.82 \]

Step 7: Calculate Exact Year

Since \( t \) is approximately 16.82, the population is expected to reach 45,000 in the year 2000 + 17 = 2017.

Key Concepts

Population Prediction

Predicting a city's population involves understanding how it changes over time. In our example, the city's population is growing linearly, meaning it increases by a constant number of people each year. We can use this information to predict future populations by applying a straightforward method.

• Identify the initial population size. Here, it was 30,700 in the year 2000.
• Determine the constant growth rate, which was 850 people per year.
• Create a linear function to represent population growth over time.

With these steps, you can predict the population for a given year, such as 2010, or determine when the population will reach a specific number, like 45,000. This approach emphasizes how the population evolves based on simple arithmetic increases.
Such predictions are useful for city planning, allocating resources, and setting policies to manage future needs.

Growth Rate

The growth rate is a crucial component of predicting future populations! It tells us how quickly the population is increasing. In linear growth scenarios like this one, the growth rate is a constant value, representing the same number of new residents each year.

• In the exercise, the growth rate was 850 people per year.
• This means every year, exactly 850 more people are added to the population.

Importance of the Growth Rate

The growth rate is vital for making accurate long-term predictions. By understanding this constant rate, we can calculate the future population during any given year by simply multiplying the growth rate by the number of years that have passed since the starting point. This provides a clear picture of population trends and potential future challenges the city may face, such as increased demand for infrastructure or services.

Mathematical Function

In population studies, a mathematical function helps us model how things change over time. Here, a linear equation was used to represent the population of the city as it grows.

• The initial population in 2000 was the starting point of this function.
• Each year adds the growth rate multiplied by the number of years since 2000.

Understanding the Linear Function

For this exercise, the function given is:\[ P(t) = 30700 + 850t \]where:

• \( P(t) \) stands for the population at the time \( t \)
• \( 30700 \) is the initial population
• \( 850 \cdot t \) represents the additional population based on the growth rate and years passed

Linear functions are simple yet powerful tools in mathematical modeling. They provide a straightforward way to calculate outcomes and predict future events based on initial conditions and constant rates of change.