Problem 81 - Precalculus

Question

The U.S. Fish and Wildlife Service is releasing a population of the endangered Mexican gray wolf in a protected area along the New Mexico and Arizona border. They estimate the population of the Mexican gray wolf to be approximated by $$ P(t)=\frac{200}{1+24 e^{-0.2 t}} $$ How many years will it take for the population to reach 100 wolves?

Step-by-Step Solution

Step 1: Apply trigonometric identities

Use the relevant trigonometric identities or formulas to simplify or evaluate the expression.

Step 2: Compute the result

Evaluate using the identities and simplify.

Key Concepts

Population Modeling

Population modeling is a crucial tool used to understand how populations grow or decline over time. It helps ecologists, zoologists, and conservationists predict future changes in a population based on current data. In our exercise, we are looking at the population of the endangered Mexican gray wolf, which is modeled using a specific mathematical formula. This formula accounts for factors such as initial population size and growth rate.

The formula given, \( P(t) = \frac{200}{1 + 24e^{-0.2t}} \), is a classic example of a logistic growth model. In population modeling, logistic models are used when there is a carrying capacity, reflecting the maximum population size the environment can sustain. This approach helps in understanding how quickly a population can grow before resources like space and food become limiting factors.

• In this specific equation, 200 represents the maximum population size or carrying capacity.
• The expression \( 24e^{-0.2t} \) is related to the speed of population growth and how it slows down as it approaches the carrying capacity.

Understanding these elements of the equation is essential for interpreting how models predict population changes and how interventions can be planned to protect endangered species.

Solving Equations

Once the model equation is set up, the next step is solving it to find when a specific condition, such as a population size of 100 wolves, will occur. For the Mexican gray wolf exercise, we have to solve \(\frac{200}{1 + 24e^{-0.2t}} = 100\) for the variable \( t \).

To solve this equation, we start by isolating the exponential term. Begin by cross-multiplying to eliminate the fraction:
\[ 200 = 100(1 + 24e^{-0.2t}) \]
Simplify the equation:
\[ 200 = 100 + 2400e^{-0.2t} \]
Next, subtract 100 from both sides:
\[ 100 = 2400e^{-0.2t} \]
Then, divide both sides by 2400 to isolate the exponential term:
\[ \frac{100}{2400} = e^{-0.2t} \]
By following these steps, we can further simplify until we use logarithms to solve for \( t \). Understanding how to solve such equations is fundamental in applying mathematical models to real-world situations.

Logarithmic Functions

Logarithmic functions are vital in solving equations that contain exponential terms. Once we simplify the model equation for the Mexican gray wolf population, we use logarithms to solve for the time \( t \).

Continuing from the previous steps, we reached an equation involving an exponential function:
\[ \frac{1}{24} = e^{-0.2t} \]
To solve for \( t \), take the natural logarithm (ln) of both sides:
\[ \ln\left(\frac{1}{24}\right) = \ln(e^{-0.2t}) \]
The logarithmic identity \( \ln(e^x) = x \) simplifies the right side to \(-0.2t\):
\[ \ln\left(\frac{1}{24}\right) = -0.2t \]
Finally, solve for \( t \) by dividing both sides by \(-0.2\):
\[ t = \frac{\ln\left(\frac{1}{24}\right)}{-0.2} \]
Using logarithms in this way allows us to find the exact time it will take for the population to reach the desired number. This is a powerful technique used extensively in exponential growth and decay problems.