Problem 67 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the indicated value, and graph the situation showing the solution point. The population of a small town is modeled by the equation \(P=1650 e^{0.5 t}\) where \(t\) is measured in years. In approximately how many years will the town's population reach \(20,000 ?\)

Step-by-Step Solution

Answer

The town's population will reach 20,000 in approximately 5 years.

Step 1: Write the Given Equation

The population model is given by the equation \(P = 1650 e^{0.5 t}\). We are tasked with finding the time \(t\) when the population \(P\) reaches 20,000.

Step 2: Set Up the Equation to Solve for t

Substitute \(P = 20,000\) into the equation: \(20,000 = 1650 e^{0.5 t}\). We need to solve this equation for \(t\).

Step 3: Isolate the Exponential Term

Divide both sides of the equation by 1650 to isolate the exponential term: \(\frac{20,000}{1650} = e^{0.5 t}\).

Step 4: Simplify the Equation

Calculate \(\frac{20,000}{1650}\) to get approximately: \(12.1212 = e^{0.5 t}\).

Step 5: Apply the Natural Logarithm

Take the natural logarithm on both sides of the equation to eliminate the exponential base: \(\ln(12.1212) = \ln(e^{0.5 t})\).

Step 6: Solve for t

Since \(\ln(e^{0.5 t}) = 0.5 t\), we get \(0.5 t = \ln(12.1212)\). Calculate \(\ln(12.1212)\) and then divide by 0.5 to find \(t\).

Step 7: Calculate the Value of t

Perform the calculation: \(t = \frac{\ln(12.1212)}{0.5}\). Using a calculator, you'll find that \(\ln(12.1212) \approx 2.495\), so \(t \approx \frac{2.495}{0.5} \approx 4.99\).

Step 8: Conclusion and Graphing

Thus, it will take approximately 5 years for the population to reach 20,000. Plotting the graph, use the point \((5, 20000)\) as the solution point.

Key Concepts

population modeling

Population modeling is an essential tool in understanding how populations change over time. It helps predict future population sizes using mathematical functions based on current data.
To model population growth, we often use an exponential function due to its simplicity and effectiveness for certain populations where the rate of growth is proportional to the current population size.

• Exponential growth occurs when the growth rate of a population is proportional to the existing population.
• This model applies to populations with abundant resources and minimal environmental constraints.
• The general form of an exponential growth model is given by the equation: \[ P(t) = P_0 e^{rt} \] where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, \( e \) is the base of the natural logarithm, and \( r \) is the rate of growth.

Understanding population modeling is crucial for making informed decisions about resource allocation, urban planning, and environmental conservation.

natural logarithm

The natural logarithm is a special logarithm that uses the base \( e \), a mathematical constant approximately equal to 2.71828. It's a valuable tool in mathematics, particularly when dealing with exponential functions.

• The natural logarithm, denoted as \( \ln(x) \), converts the exponential term into a linear one, allowing easier manipulation and solution of equations.
• When solving problems involving exponential growth, the natural logarithm helps isolate the variable of interest, such as time \( t \) in population growth models.
• Using \( \ln \) allows us to reframe an equation like \( e^{x} = a \) into the more manageable form of \( x = \ln(a) \).

This function is pivotal in calculus and complex calculations, especially when working with natural growth processes represented by exponential functions.

solving exponential equations

Exponential equations can be challenging due to the presence of variables in exponents. However, with a systematic approach, they become manageable and solvable. Here's how we tackle them:1. **Isolate the Exponential Expression**: Start by isolating the exponential term on one side of the equation. For example, with \( 20,000 = 1650 e^{0.5t} \), dividing both sides by 1650 sets it up as \( e^{0.5t} = 12.1212 \).
2. **Apply the Natural Logarithm**: Use the natural logarithm to eliminate the exponential base, allowing the equation to be written in a simpler form, such as \( \ln(e^{0.5t}) = \ln(12.1212) \).
3. **Solve for the Variable**: Simplify using the property \( \ln(e^{x}) = x \), leading to \( 0.5t = \ln(12.1212) \). Solve for \( t \) by rearranging the equation and simplifying.These techniques transform exponential equations into linear ones, providing a straightforward path to a solution.

graphing solutions

After solving an exponential equation, visualizing the solution on a graph provides a clear understanding of the behavior of the equation and its implications in real-world scenarios. Graphing allows us to see trends and verify solutions. Here’s how to graph an exponential solution:

• **Determine Key Points**: Identify significant points from the equation, like the initial population and the solution point. For instance, plot the initial population \( 1650 \) when \( t = 0 \) and the solution point \( (5, 20000) \).
• **Draw the Curve**: Use these points to sketch the exponential curve. The graph should rise steeply, reflecting exponential growth.
• **Verify the Solution**: Check that the curve passes through your calculated solution point, confirming its accuracy.

Graphing not only verifies computational results but also aids in interpreting data, making it a valuable step in understanding exponential growth scenarios.