Problem 32 - A Graphical Approach to College Algebra

Question

Solve each problem. Bicycle Safety \(\quad\) A survey found that \(76 \%\) of bicycle riders do not wear helmets. (Source: Opinion Research Corporation for Glaxo Wellcome, Inc.) (a) Write a linear function \(f\) that computes the number of people who do not wear helmets among \(x\) bicycle riders. (b) There are approximately 38.7 million riders of all ages who do not wear helmets. Write a linear equation whose solution gives the total number of bicycle riders. Find this number of riders.

Step-by-Step Solution

Answer

(a) \( f(x) = 0.76x \); (b) Total riders \( \approx 50.92 \) million.

Step 1: Define the Function

First, we need to determine the linear function that models the number of riders not wearing helmets. We have a percentage (76%) of riders who do not wear helmets. Since a linear function is usually of the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants, and \( b = 0 \) because when there are no riders, no one wears helmets. Here, \( m \) is the fraction of riders not wearing helmets, which is 0.76. Therefore, the function is \( f(x) = 0.76x \).

Step 2: Use the Function to Determine Total Riders

Now, we use the linear function to find the total number of bicycle riders, given that approximately 38.7 million riders do not wear helmets. We set \( f(x) \) to 38.7 million in the equation \( f(x) = 0.76x \). Thus, we solve \( 0.76x = 38.7 \) million.

Step 3: Solve for \( x \)

To solve for \( x \), divide both sides of the equation by 0.76: \[ x = \frac{38.7}{0.76} \]. Calculate this value to find the total number of bicycle riders.

Step 4: Calculate the Exact Number

Divide 38.7 by 0.76 to get the value of \( x \). Performing the division gives \( x \approx 50.92 \) million. Therefore, the approximate total number of bicycle riders is 50.92 million.

Key Concepts

Percentage

Understanding percentages is an integral part of solving many mathematical problems, including linear functions. In this exercise, we are given that 76% of bicycle riders do not wear helmets.
This percentage can be thought of as a part per 100, meaning that out of every 100 riders, 76 do not wear helmets.

When faced with such problems, it is helpful to convert percentages to decimals for easy calculations. This is done by dividing the percentage by 100. So, 76% becomes 0.76.

• Percentage to Decimal: To convert, divide by 100.


• Decimal Form: Fraction of people not wearing helmets is 0.76.

This decimal representation allows us to construct a linear function to model the situation accurately.

Equation Solving

Solving equations is a central aspect of algebra, and it requires a systematic approach. In this exercise, we use equation solving to find the total number of bicycle riders based on the known number of riders not wearing helmets.
The equation we have is derived from the linear function: \[f(x) = 0.76x = 38.7\] This equation sets up a direct relationship between the total number of riders, represented by \(x\), and the riders who don’t wear helmets.

To solve for \(x\), the following steps are used:

• Identify the equation: \(0.76x = 38.7\) million.


• Isolate \(x\): Divide both sides by 0.76.


• Calculation: \(x = \frac{38.7}{0.76} \approx 50.92\) million.

By dividing 38.7 million by 0.76, we determine that there are approximately 50.92 million total bicycle riders. This solution showcases the power of linear equations to solve real-world problems.

Mathematical Modeling

Mathematical modeling is an essential skill in representing real-world scenarios using mathematical concepts. In this exercise, we model the relation between bicycle riders and helmet usage with a linear function.
This is what makes a mathematical model - it turns a complex real-world situation into something understandable and solvable through math. In simpler terms, modeling helps us see the structure.

Steps to create a model in this context include:

• Define Variables: Let \(x\) be the total riders.


• Use Known Data: 76% don’t wear helmets – given as a decimal (0.76).
• Linear Function: \(f(x) = 0.76x\), representing the model.

Through this linear function, we quickly calculate estimates and make business sense from the data. Mathematical modeling thus plays a crucial role in making sense of surveys and planning safety interventions.