Question

A fish farm plans to expand. The fish population, \(P,\) in hundreds of thousands, as a function of time, \(t,\) in years, can be modelled by the function \(P(t)=6(1.03)^{t}\) The farm biologists use the function \(F(t)=8+0.04 t,\) where \(F\) is the amount of food, in units, that can sustain the fish population for 1 year. One unit can sustain one fish for 1 year. a) Graph \(P(t)\) and \(F(t)\) on the same set of axes and describe the trends. b) The amount of food per fish is calculated using \(y=\frac{F(t)}{P(t)} .\) Graph \(y=\frac{F(t)}{P(t)}\) on a different set of axes. Identify a suitable window setting for your graph. Are there values that should not be considered? c) At what time is the amount of food per fish a maximum?d) The fish farm will no longer be viable when there is not enough food to sustain the population. When will this occur? Explain how you determined your result.

Short answer

Graph \( P(t) \) and \( F(t) \), describe, find \( y(t) \), graph y. Maximum food per fish: use calculus to find peak \( y(t) \. Non-viable point: solve \( F(t) < P(t) \).

Step-by-step solution

Understanding the functions

Given two functions: \(P(t) = 6(1.03)^t\) representing the fish population in hundred thousands and \(F(t) = 8 + 0.04t\) representing the amount of food in units. Understand that the fish population grows exponentially while the food grows linearly.

Graphing the functions

Graph the functions \(P(t) = 6(1.03)^t\) and \(F(t) = 8 + 0.04t\) on the same axes. On a graphing calculator or software, set an appropriate window to capture the trends clearly. Typically, a window with the x-axis (time, t) ranging from 0 to 50 years and the y-axis (value of P and F) from 0 to 20 units will be useful. Note the exponential growth of P(t) compared to the linear growth of F(t).

Describe the trends

As time progresses, \(P(t) = 6(1.03)^t\) increases exponentially, meaning the fish population rapidly grows after some years. In contrast, \(F(t) = 8 + 0.04t\) increases linearly, indicating a gradual increase in the amount of food available. Therefore, at some point, the fish population will outgrow the food supply.

Define the food per fish function

The amount of food per fish can be defined by the function \(y = \frac{F(t)}{P(t)}\). Express it as: \[y(t) = \frac{8 + 0.04t}{6(1.03)^t}\].

Graphing y(t)

Graph the function \(y(t) = \frac{8 + 0.04t}{6(1.03)^t}\) on a new set of axes. Choose a suitable window setting; a possible one might be 0 to 50 years on the x-axis and 0 to 1 units on the y-axis. Examine the behavior and identify asymptotic behavior where y approaches zero.

Identify suitable and unsuitable values

Examine the graph and identify time values where y(t) becomes too small, indicating insufficient food. Inappropriate values of t (time) for the function could be large values where y(t) is approaching zero.

Determine maximum food per fish

To find where the food per fish is maximum, observe the graph of \(y(t)\). The maximum point occurs at the peak of the graph. Use calculus to find the critical point: set the derivative \(y'(t) = 0\). Compute \[y'(t) = \frac{d}{dt}\left( \frac{8 + 0.04t}{6(1.03)^t} \right)\], set \[\frac{d}{dt}\] to 0 and solve for \(t\).

Determine when the farm becomes non-viable

Establish when the food supply will no longer sustain the fish population by finding when \(F(t) < P(t)\). Solve for \(t\) in the inequality \(8 + 0.04t < 6(1.03)^t\). This will indicate the non-viable point when the fish population exceeds the available food supply.

Key concepts

Exponential Functions

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. In the context of our exercise, the function representing the fish population over time is given by: \( P(t) = 6(1.03)^t \). This means that the fish population starts at 600,000 (since 6 represents 6 hundred thousand) and increases by 3% each year. Exponential growth is characterized by this rapid increase, where each step grows by a fixed percentage rather than a fixed amount.
Some key points about exponential functions:

• The base must be greater than 1 for positive growth.
• Exponential functions grow faster than linear functions as time increases.
• Population growth, radioactive decay, and interest calculations often use exponential models.

The takeaway message here is that exponential growth can lead to large quantities quickly, which is important for understanding how the fish population expands exponentially over time in this problem.

Linear Growth

Linear growth refers to an increase by the same amount in each time period. This is represented in our exercise by the function for food supply: \(F(t) = 8 + 0.04t\). Here, the food starts at 8 units and increases by 0.04 units each year. Unlike exponential growth, linear growth is steady and predictable.
Some essential features of linear growth:

• A linear function creates a straight-line graph.
• The slope represents the rate of growth or decline.
• In this specific example, the slope is 0.04, meaning that the food amount increases by 0.04 units per year.

While linear growth is simpler and easier to predict, it generally accumulates resources or values at a much slower rate compared to exponential growth. This distinction is critical in our model, where the food supply fails to keep up with the rapidly growing fish population over time.

Population Modeling

Population modeling involves using mathematical functions to represent changes in population over time. In this exercise, we use exponential and linear functions to model the fish population and the food supply, respectively. This combination helps us analyze and predict future conditions for the fish farm.
Key elements of population modeling include:

• Variables: These represent quantities such as the population size (\(P(t)\)) and food supply (\(F(t)\)).
• Time: The variable over which changes are tracked, in this case, years (\(t)\)).
• Growth Rate: The rate at which the population or resource increases (3% exponential for fish, 0.04 units per year linear for food).

By combining these elements, we can form equations that predict how the fish population and food supply will change over time. For example, the inequality \(F(t) < P(t)\) helps us determine when there will be insufficient food to sustain the fish population, which is when the farm will become non-viable. Understanding these models allows us to make informed decisions about resource allocation and management for sustainable practices.