Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The length (in centimeters) of a typical Pacific halibut \(t\) yr old is approximately $$ f(t)=200\left(1-0.956 e^{-0.18 t}\right) $$ What is the length of a typical 5 -yr-old Pacific halibut?

Step-by-Step Solution

Answer

The length of a typical 5-year-old Pacific halibut is approximately 76.493 centimeters.

Step 1: Identify the given function for the Pacific halibut's length

We are given the function f(t) which represents the length of a Pacific halibut of age t years old: \( f(t)=200\left(1-0.956 e^{-0.18 t}\right) \)

Step 2: Plug in the given age (5 years) into the formula

We are asked for the length of a 5-year-old Pacific halibut. We need to replace t with 5 in the function: \( f(5)=200\left(1-0.956 e^{-0.18(5)}\right) \)

Step 3: Simplify the function

Calculate the exponential term and multiplications inside the parentheses first: \( f(5)=200\left(1-0.956 e^{-0.9}\right) \)

Step 4: Continue simplifying and find the length

Finish the calculations in the function to find the length of a 5-year-old Pacific halibut: \( f(5)=200\left(1-0.956 e^{-0.9}\right) \approx 76.493 \) So, the length of a typical 5-year-old Pacific halibut is approximately 76.493 centimeters.

Key Concepts

Mathematical Modeling

Mathematical modeling is like creating a blueprint using mathematics. In the world of fish, such as the Pacific halibut, it's a tool to describe and predict how these creatures grow over time. Here, the mathematical model is expressed through the function \(f(t) = 200\left(1 - 0.956 e^{-0.18 t}\right)\). This formula is powerful because it helps us understand the growth pattern of halibuts without measuring them physically.

• We represent real-world matters with equations, making complex things simple.
• The function uses variables and constants to map the relationship between age and length.

In this context, mathematical modeling allows scientists and researchers to make informed decisions and predictions about fish populations, supporting conservation and fishing industries by giving insight into future sizes and weights.

Growth Functions

Growth functions like the one used for the halibut are essential for predicting changes over time. This function, \(200\left(1-0.956 e^{-0.18 t}\right)\), shows how the halibut's length changes with age.

• Exponential components (\(e^{-0.18 t}\)) depict how growth rates decrease over time.
• Constants like 200, and multipliers such as 0.956, adjust the scale and rate of this growth.

Such functions are commonly seen in areas like ecology, where they help model population dynamics. They are not linear, meaning as fish age, their growth rate slows down. By understanding this function, you can see that the halibut's length approaches 200 cm, indicating a growth limit.
This approach also applies to other scenarios beyond biology, like financial growth and radioactive decay, illustrating their versatility and importance.

Problem Solving

Solving real-world problems using math involves breaking them down into simpler parts. When determining the length of the 5-year-old halibut, the process was straightforward yet informative.

• Start by identifying the function we need: \(f(t)=200\left(1-0.956 e^{-0.18 t}\right)\).
• Replace the variable \(t\) with the specific age to analyze, like \(t=5\) in this case.
• Step by step, simplify the exponential term and then the arithmetic calculations to find the answer.

Approaching problems systematically helps in understanding complex problems with ease.
This step-by-step method breaks down larger tasks into smaller, manageable ones, making problem-solving in mathematics less daunting and more engaging.