15–20. Designing exponential growth functions Complete the following steps for the given situation.


a. Find the rate constant k and use it to devise an exponential growth function that fits the given data.
b. Answer the accompanying question.


Population The population of Clark County, Nevada, was about 2.115 million in 2015. Assuming an annual growth rate of 1.5%/yr, what will the county population be in 2025?

Verified step by step guidance

1. Step 1: Identify the given information and variables. The initial population at time \(t=0\) (year 2015) is \(P_0 = 2.115\) million. The annual growth rate is 1.5%, which can be written as a decimal \(r = 0.015\). The time elapsed from 2015 to 2025 is \(t = 10\) years.
2. Step 2: Recall the general form of the exponential growth function: \(P(t) = P_0 \cdot e^{k t}\), where \(k\) is the rate constant we need to find.
3. Step 3: Relate the given annual growth rate to the rate constant \(k\). Since the population grows by 1.5% per year, the growth factor per year is \(1 + r = 1.015\). This corresponds to \(e^{k} = 1.015\), so solve for \(k\) by taking the natural logarithm: \(k = \ln(1.015)\).
4. Step 4: Write the exponential growth function using the calculated \(k\): \(P(t) = 2.115 \cdot e^{k t}\), where \(k = \ln(1.015)\) and \(t\) is the number of years after 2015.
5. Step 5: To find the population in 2025, substitute \(t = 10\) into the function: \(P(10) = 2.115 \cdot e^{k \cdot 10}\). This expression gives the population in millions at the year 2025.

Key Concepts

Exponential Growth Function

An exponential growth function models quantities that increase at a rate proportional to their current value. It is generally expressed as P(t) = P_0 * e^(kt), where P_0 is the initial amount, k is the growth rate constant, and t is time. This function is essential for predicting population changes over time.

Rate Constant (k) in Exponential Growth

The rate constant k represents the continuous growth rate in the exponential model. It can be found by converting a given percentage growth rate to a decimal and using the relationship k = ln(1 + r), where r is the growth rate per time unit. Determining k allows accurate formulation of the growth function.

Applying the Exponential Model to Population Prediction

To predict future population, substitute the initial population, the rate constant k, and the elapsed time into the exponential growth formula. This calculation estimates the population at a future date, assuming the growth rate remains constant, which is crucial for planning and analysis.