Problem 93 - Intermediate Algebra

Question

National Parks. The number of visitors \(v(t)\) to U.S. national parks can be approximated by the function \(v(t)=-100,000 t+3,600,000,\) where \(t\) is the number of years after \(1990 .\) Use an inequality to determine those years for which the number of visitors fell below \(2,400,000 .\) (Source: National Park Service Stats)

Step-by-Step Solution

Answer

The number of visitors fell below 2,400,000 after the year 2002.

Step 1: Set Up the Inequality

We need to find those years where the function for visitors \( v(t) = -100,000t + 3,600,000 \) is less than \( 2,400,000 \). Set up the inequality: \( -100,000t + 3,600,000 < 2,400,000 \).

Step 2: Simplify the Inequality

Subtract \( 3,600,000 \) from both sides of the inequality to isolate the term with \( t \): \( -100,000t < 2,400,000 - 3,600,000 \). This simplifies to \( -100,000t < -1,200,000 \).

Step 3: Solve for \( t \)

Divide both sides by \( -100,000 \). Remember that dividing by a negative number reverses the inequality sign. So, we have \( t > \frac{-1,200,000}{-100,000} \). This simplifies to \( t > 12 \).

Step 4: Identify the Year

Since \( t \) is the number of years after 1990, we add 1990 to \( 12 \) to find the specific year. So, \( 1990 + 12 = 2002 \). Therefore, the number of visitors fell below 2,400,000 in the year \( 2002 \).

Key Concepts

Function Approximation

Function approximation is a mathematical concept used to estimate the values of a complex function with a simpler one. This technique is widely used in various fields, including statistics, engineering, and economics, to model real-world situations. In the context of our exercise, the function \( v(t) = -100,000t + 3,600,000 \) serves as an approximate model for the number of visitors to U.S. national parks each year. Here, the variable \( t \) represents years after 1990.

• \(-100,000t\): This term indicates how the number of visitors is expected to decline each year.
• \(3,600,000\): This is a constant that sets the starting point, representing the estimated visitors in 1990.

Understanding function approximation helps us translate complex data into manageable models to predict future trends with reasonable accuracy.

Linear Equations

Linear equations represent a straight line and are one of the simplest forms of mathematical expressions. They are represented in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In our exercise, the given function \(v(t) = -100,000t + 3,600,000\) is a linear equation:

• The slope (\(-100,000\)) shows how the number of visitors changes each year.
• The y-intercept (\(3,600,000\)) indicates the initial point when \( t = 0 \), which corresponds to the year 1990.

By understanding this equation, we can see how the number of visitors is expected to change linearly over time, allowing us to solve problems regarding visitor trends.

Problem-Solving Steps

Solving equations and inequalities involves a series of systematic problem-solving steps. Our exercise uses these steps to find when visitor numbers fall below a certain level:- **Set Up the Inequality:** Translate the problem into a mathematical inequality \(-100,000t + 3,600,000 < 2,400,000\).- **Simplify the Inequality:** Eliminate complex parts by subtracting or adding terms on both sides. This simplifies our inequality to \(-100,000t < -1,200,000\).- **Solve for Unknown:** Solve for \(t\) by isolating it. Divide both sides by \(-100,000\), remembering to reverse the inequality sign. Thus, we find \(t > 12\).- **Interpret the Result:** Convert \(t\) into a real-world year by adding it to the base year, 1990.Each problem-solving step builds on the prior one, ensuring we maintain logical consistency while solving the problem.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world phenomena to better understand and predict behaviors. In this context, the function \(v(t) = -100,000t + 3,600,000\) is a model predicting visitor numbers over time.
To use mathematical modeling effectively:

• Identify the problem: Understand what real-world situation you are trying to model.
• Formulate: Translate it into mathematical language, like an equation or inequality.
• Analyze: Use mathematical techniques to explore predictions and insights.
• Interpret: Convert mathematical results back into real-world context.

In our exercise, modeling helps us understand visitor trends and make informed decisions about resource allocation and management within national parks.