Problem 72 - Precalculus : Building Concepts and Connections

Question

You will use polynomial functions to study real-world problems. The number of species on the U.S. endangered species list during the years \(1998-2005\) can be modeled by the function $$f(t)=0.308 t^{3}-5.20 t^{2}+32.2 t+921$$ where \(t\) is the number of years since \(1998 .\) (Source: U.S. Fish and Wildlife Service) (a) Find and interpret \(f(0)\). (b) How many species were on the list in \(2004 ?\) (c) \(\quad\) Use a graphing utility to graph this function for \(0 \leq t \leq 7 .\) Judging by the trend seen in the graph, is this model reliable for long- term predictions? Why or why not?

Step-by-Step Solution

Answer

a) In 1998, there were 921 species on the U.S. endangered species list. b) In 2004, there were approximately 1063 species on the list. c) The model may not be reliable for long-term predictions because it predicts a decrease in the number of endangered species after a certain time.

Step 1: Find and interpret \(f(0)\)

By substituting \(t = 0\) into the function, we find \(f(0) = 0.308(0)^3 - 5.20(0)^2 + 32.2(0) + 921 = 921\). This represents the number of species on the U.S. endangered species list in the year 1998.

Step 2: Find the number of species on the list in 2004

Here, \(t\) represents the number of years since 1998. So, for the year 2004, \(t = 2004 - 1998 = 6\). Substituting \(t = 6\) into the function, we find \(f(6) = 0.308(6)^3 - 5.20(6)^2 + 32.2(6) + 921\), which equals approximately 1063. This suggests that in 2004, there were approximately 1063 species on the U.S. endangered species list.

Step 3: Graph and Interpret the function

Use a graphing utility to plot the function for values of \(0 \leq t \leq 7\). The graph will show a curve that initially slopes upwards, indicating an increase in the number of endangered species over the years. Whether this model is reliable for long-term predictions would depend on whether this trend is expected to continue. Given that the function is cubic and starts to curve downwards for larger \(t\), it suggests that the model may not be reliable for long-term predictions because it predicts a decrease in the number of endangered species after a certain time.

Key Concepts

Endangered Species Modeling

Modeling the number of endangered species with a polynomial function can help us understand trends over time. The given function represents the number of species on the U.S. endangered list between 1998 and 2005. By analyzing this model, we gain insight into how the number changed over these years. The model captures the dynamics over a short period, but it does not necessarily predict future trends accurately due to its complexity and the nature of endangered species, which can vary with environmental and legislative changes. It is crucial to continuously update models with new data to maintain accuracy.

Cubic Functions

Cubic functions are polynomial functions of degree three. They have the form \( f(x) = ax^3 + bx^2 + cx + d \)where \( a, b, c, \) and \( d \) are constants. They can model more complex, non-linear relationships than linear or quadratic functions because they can change direction twice.

• The graph of a cubic function may have one or two bends, known as turning points.
• In some cases, cubic functions can describe systems where the rate of change itself changes over time.

The function in this exercise is cubic, allowing it to model the increase in endangered species, as well as potential decreases later. This is due to its ability to curve upwards and then downwards.

Graphing Utilities

Graphing utilities are essential tools that help us visualize mathematical functions. They are particularly useful for functions like cubic ones, where more complex curves need to be evaluated. To graph the function for the given range:

• Enter the function into a graphing calculator or software.
• Set the range for \( t \) from 0 to 7 since it represents years since 1998.

A well-drawn graph helps in understanding the trends of endangered species over time and assessing the reliability of the function for future predictions.

Mathematical Interpretation

Interpreting mathematical models involves analyzing the results they produce. When given a polynomial like the one in this exercise, substituting values for \( t \) gives us the number of endangered species for specific years.

• For example, \( f(0) = 921 \) means 921 species were listed in 1998.
• Plugging \( t = 6 \) into the function estimates 1063 species in 2004.

The interpretation also extends to understanding the reliability of the model. By graphing the function, we can see how predictions might deviate if it is used beyond the intended timeframe. Polynomial models can show long-term trends, making them beneficial, but they are only as reliable as the data and assumptions on which they are based.