Problem 2 - Algebra and Trigonometry

Question

The number of a certain species of fish is modeled by the function $$n(t)=12 e^{0.012 t}$$ where \(t\) is measured in years and \(n(t)\) is measured in millions. (a) What is the relative rate of growth of the fish population? Express your answer as a percentage. (b) What will the fish population be after 5 years? (c) After how many years will the number of fish reach 30 million? (d) Sketch a graph of the fish population function \(n(t) .\)

Step-by-Step Solution

Answer

(a) 1.2% (b) 12.74 million (c) 76.42 years.

Step 1: Find the Relative Rate of Growth

The relative rate of growth of a population modeled by an exponential function of the form \( n(t) = n_0 e^{rt} \) is given by the constant \( r \). In our function, \( n(t) = 12 e^{0.012t} \), the constant \( r \) is \( 0.012 \). To express as a percentage, simply multiply by 100. Thus, the relative rate of growth is \( 0.012 \times 100 = 1.2\% \).

Step 2: Calculate the Population After 5 Years

To find the fish population after 5 years, substitute \( t = 5 \) into the function \( n(t) = 12 e^{0.012t} \).\[n(5) = 12 e^{0.012 \times 5}\]Calculating inside the exponential first:\[ e^{0.06} \approx 1.06183655 \]Thus,\[ n(5) = 12 \times 1.06183655 \approx 12.7420382 \]Therefore, the population after 5 years is approximately 12.74 million.

Step 3: Determine Years to Reach 30 Million

We need to solve for \( t \) such that \( n(t) = 30 \). Start with the equation:\[ 12 e^{0.012t} = 30 \]Divide both sides by 12:\[ e^{0.012t} = 2.5 \]Take the natural logarithm of both sides to solve for \( t \):\[ 0.012t = \ln(2.5) \]\[ t = \frac{\ln(2.5)}{0.012} \approx 76.42 \]Therefore, it will take approximately 76.42 years for the population to reach 30 million.

Step 4: Sketch the Graph of the Population Function

To sketch the graph of the function \( n(t) = 12 e^{0.012t} \), recognize that it is an exponential growth function. It starts at 12 when \( t = 0 \) (since \( e^0 = 1 \)), and grows exponentially without bounds as \( t \) increases. The curve will gradually rise from the point (0, 12) and will steepen its ascent as time progresses, illustrating the continuous and increasing growth of the fish population.

Key Concepts

Relative Rate of Growth

The relative rate of growth is a key concept when understanding how populations change over time.
In the context of our fish population example, the relative rate of growth informs us about the speed at which the population increases annually. It is represented by the constant \( r \) in the exponential function, which is given by the equation: \[ n(t) = n_0 e^{rt} \].
This constant value, \( r \), shows how the population grows per unit of time (years in this case).

• In our example, \( r = 0.012 \), which translates to a relative growth rate of \( 1.2\% \) (as you multiply \( 0.012 \) by 100 to convert it to a percentage).

This percentage tells you that each year the population is expected to grow by an additional 1.2% of its size the previous year. This understanding is crucial in predicting and planning for changes in wildlife populations, especially for ecosystem management and conservation efforts.

Exponential Function

An exponential function is a mathematical function that describes situations where a quantity grows or decays at a rate proportional to its current value.
The formula is generally expressed as \( n(t) = n_0 e^{rt} \), where:

• \( n(t) \) represents the quantity or size of what we are measuring at time \( t \).
• \( n_0 \) is the initial quantity/amount when \( t = 0 \).
• \( e \) is the base of the natural logarithm, approximately equal to \( 2.718 \).
• \( r \) is the rate of growth, which is constant.
• \( t \) stands for time or the independent variable.

In our fish population function, \( 12 e^{0.012t} \), this pattern of exponential growth helps in predicting future population sizes after a specific duration. Such functions are pivotal in many fields, including biology, finance, and physics, offering a means to model growth processes effectively over time.

Population Modeling

Population modeling with exponential functions offers insights into future population dynamics based on existing data.
It helps in understanding how a particular species will grow under certain conditions. With respect to the fish population function provided:

• \( n(t) = 12 e^{0.012t} \) models the change in the population in millions over time.
• It allows us to find values at specific times, like \( n(5) \) for the population in 5 years.
• Also determines when the population will reach a certain size, such as finding the needed years to reach 30 million.

Population modeling is a powerful tool that offers more than just numbers; it aids in decision-making for environmental management and conservation strategies.
By anticipating changes in populations, effective measures can be implemented to ensure balance and sustainability in ecosystems.