Answered

The population of a certain fish species can be modeled using the equation

$P(t)=50 e^{0.15 t}$

where $t$ is the number of years after 2018.

A particular lake can accommodate at most 3,500 of this fish species.



Based on the model, during which year will the population of fish first exceed the capacity of the lake?


A2022
B2028
C2046
D2088

Verified Answer :

  • The population model is \(P(t) = 50e^{0.15t}\), and we set it equal to the limit of 3,500.
  • Dividing by 50 and taking the natural logarithm leads to \(0.15t > \ln(70)\).
  • Solving for \(t\) gives \(t > \frac{\ln(70)}{0.15} \approx 28.32\).
  • Adding 28.32 years to 2018 points to the year 2046.
  • The correct option is \(\boxed{2046}\).

Explanation

  1. Understanding the Problem and Data Analysis

The problem asks us to determine when the population of a specific fish species, modeled by the function \(P(t) = 50e^{0.15t}\), will exceed the capacity of a lake, which is \(3,500\) fish. Here, \(t\) denotes the number of years after 2018. To find the time \(t\) when the population first exceeds the capacity, we set up the inequality \(P(t) > 3,500\).

  1. Solving for Time \(t\)

We begin by writing the inequality using the given model:
\[50e^{0.15t} > 3,500\]
Next, we isolate the exponential term by dividing both sides by 50:
\[e^{0.15t} > \frac{3,500}{50}\]
\[e^{0.15t} > 70\]
To solve for \(t\), we take the natural logarithm (\(\ln\)) of both sides:
\[0.15t > \ln(70)\]
Dividing by 0.15, we get:
\[t > \frac{\ln(70)}{0.15}\]
Using the approximate value of \(\ln(70) \approx 4.2485\), we find:
\[t > \frac{4.2485}{0.15} \approx 28.32\]

  1. Determining the Year and Final Conclusion

Since \(t\) represents the number of years after 2018, the population will exceed the lake's capacity when \(t\) is greater than approximately \(28.32\). Adding this value to the base year of 2018:
\[2018 + 28.32 = 2046.32\]
This means the population will first exceed the capacity during the year 2046. Comparing this with the given options, we select 2046 as the correct answer.

Examples

Exponential growth models are widely used in biology and ecology to predict population dynamics. For example, if a conservation agency tracks an endangered species, they might use similar growth models to determine when a sanctuary's resources—such as food or space—will reach their limits, allowing them to proactively manage the habitat or relocate individuals. Understanding how to solve these equations is essential for making data-driven environmental management decisions.

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