Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Longevity of halibut In fishery science, a cohort is the collection of fish that results from one annual reproduction. It is usually assumed that the number of fish \(N(t)\) still alive after \(t\) years is given by an exponential function. For Pacific halibut, \(N(t)=N_{0} e^{-0.2 t}\), where \(N_{0}\) is the initial size of the cohort. Approximate the percentage of the original number still alive after 10 years.

Step-by-Step Solution

Answer

13.53% of the original cohort is still alive after 10 years.

Step 1: Understanding the Problem

We need to find the percentage of the initial cohort of Pacific halibut that is still alive after 10 years using the exponential decay formula.

Step 2: Identify Given Values

The expression for the number of halibut alive after time \(t\) years is given by \(N(t) = N_0 e^{-0.2t}\). We are interested in finding \(N(10)\), the number of halibut alive after 10 years.

Step 3: Substitute the Time Value

Substitute \(t = 10\) into the expression:\[N(10) = N_0 e^{-0.2 \times 10} = N_0 e^{-2}.\]

Step 4: Calculate the Exponential Decay

Calculate \(e^{-2}\) using a calculator to find the decay factor. \(e\) is approximately 2.718, so \(e^{-2} \approx 0.1353\).

Step 5: Find the Number Alive After 10 Years

The number of halibut alive after 10 years is \(N(10) = N_0 \times 0.1353\), indicating that 13.53% of the original cohort remains.

Step 6: Calculate the Percentage Remaining

Since \(N(t)\) represents the number of fish still alive as a fraction of \(N_0\), the percentage of fish still alive is \(0.1353\times 100 = 13.53\%\).

Key Concepts

Exponential Function

An exponential function is a type of mathematical expression commonly used to model natural processes. These functions are represented as \( f(x) = a \cdot e^{bx} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.718. In these functions, \( a \) is a constant term which scales the function, and \( b \) determines the rate of growth or decay.
Exponential functions are particularly powerful in modeling scenarios where growth or decay accelerates over time. An example includes population growth, where more individuals lead to increasingly rapid growth, or decay, like radioactive material which decreases at a rate proportional to its current amount.
In fishery science, we use it to predict the number of living organisms in a group over time. As time progresses, the number typically reduces, showcasing exponential decay.
A critical takeaway is understanding that exponential decay occurs when \( b \) is negative, leading the function to shrink over time, as seen with continuous reductions in population sizes or interest.

Fisheries Science

Fisheries science is the field that focuses on understanding, managing, and conserving fish populations and habitats. It involves many disciplines such as biology, ecology, and economics, with the objective of ensuring sustainable fishing practices.
Sustainability is key in fisheries science. Scientists often engage in activities like monitoring fish populations, studying reproductive patterns, and evaluating impacts of fishing on marine ecosystems to ensure fish stocks are not depleted.
In the context of the original exercise, fisheries science helps manage fishery resources by analyzing cohorts—groups of fish born during the same breeding season. By understanding how quickly a cohort decreases, scientists can determine sustainable catch rates, ensuring enough fish survive to maintain the population.
Practitioners in fisheries science rely on data from exponential functions to make predictions about fish populations' future states, which is crucial for planning and conservation efforts.

Cohort Analysis

Cohort analysis is a vital element within fisheries science, where scientists study the behaviors and characteristics of fish groups born in the same period, called cohorts. This method is used to comprehend various lifecycle phases including birth, growth, reproduction, and death.
By observing a particular cohort over time, scientists can infer the survival rate and lifespan of a fish species. This type of analysis allows for predictions on the population's future trends and aids in sustainable management.
Using cohort analysis, scientists track how many in a specific group survive each year. For example, in the Pacific halibut example, they calculate how many fish are alive after each passing year using exponential decay functions. This offers insights into factors influencing the population and necessary adjustments in fishing policies.
Cohort analysis ensures fisheries managers maintain the health of fish populations while allowing continued fishing activities.

Percentage Calculation

Calculating the percentage is an important mathematical skill particularly useful in interpreting data, such as determining growth or decay in various contexts. It's often used to express a part of a whole as a fraction of 100. The formula is:\[ \text{Percentage} = \left( \frac{\text{part}}{\text{whole}} \right) \times 100 \]
In the context of the original exercise, we aim to find what percentage of the original cohort of Pacific halibut remains after a specific time frame.
Having calculated \( N(10) = N_0 \times 0.1353 \), where \( 0.1353 \) is the fraction of the cohort still alive after 10 years, we convert this to a percentage by multiplying by 100, resulting in 13.53%.
This percentage calculation is crucial for interpreting the exponential decay results, helping fisheries scientists understand how much of a population remains relative to its initial size at any given time. This insight is invaluable for making informed decisions on conservation and fisheries management.