Problem 30 - A Graphical Approach to College Algebra

Question

Solve each problem. Women against the Men The men's Olympic pole vaulting winning heights in meters during year \(x\) can be approximated by \(H(x)=\frac{1}{48} x-35.83,\) where \(1896 \leq x \leq 2008\) (Assume that \(x\) is a multiple of 4 because the Olympics occur every 4 years.) (a) Evaluate \(H(1920)\) and interpret the result. (b) In 2008 the women's Olympic winning height in the pole vault was about 5 meters. Determine the years when this height would have beaten or tied the men's winning heights.

Step-by-Step Solution

Answer

(a) 4.17 meters in 1920; (b) Years 1896 to 1960.

Step 1: Understanding the Function

The function given is \( H(x) = \frac{1}{48} x - 35.83 \), which estimates the men's Olympic pole vaulting winning heights in meters for any year \( x \) between 1896 and 2008. \( x \) represents the year and should be a multiple of 4 like 1896, 1900, 1904, etc.

Step 2: Evaluate H(1920)

To find \( H(1920) \), substitute 1920 for \( x \) in the formula:\[ H(1920) = \frac{1}{48} \times 1920 - 35.83 \]Calculate it:- Calculate \( \frac{1}{48} \times 1920 = 40 \).- Subtract 35.83 from 40: \( 40 - 35.83 = 4.17 \).Therefore, \( H(1920) = 4.17 \) meters. This means the winning height in 1920 was approximately 4.17 meters.

Step 3: Compare Details for Women's Winning Height

The winning height for women in 2008 is given as 5 meters. We need to find the years when \( H(x) \), the men's winning height, was less than or equal to 5 meters. This means solving \( \frac{1}{48} x - 35.83 \leq 5 \).

Step 4: Solve the Inequality for Men's Height

To determine for which years \( H(x) \leq 5 \), solve:\[ \frac{1}{48} x - 35.83 \leq 5 \]First, add 35.83 to both sides:\[ \frac{1}{48} x \leq 40.83 \]Multiply both sides by 48:\[ x \leq 1960 \]Thus, for years \( x \leq 1960 \), the men's height was less than or equal to the women's height in 2008.

Step 5: Determine Specific Years

The specific years that fit within the range \( x \leq 1960 \) with the condition that \( x \) is a multiple of 4 are 1896, 1900, 1904,...,1960. The women's height of 5 meters would have tied or beaten the men's Olympic winning heights in this range.

Key Concepts

Olympic Games Mathematics

Olympic Games Mathematics is a way to use math in understanding the historical context and changes in athletic performances. In this exercise, we are presented with a real-world mathematical model through the men's Olympic pole vault heights over time. The goal is to track performance trends and use algebraic equations simply by evaluating given formulas.
Mathematical problems like these are interesting because they reflect historical data in a format that is easy to calculate and interpret. For instance, by plugging a year into the given function, one can determine the approximate pole vault winning height for that year. This not only encourages computational skills but also helps us draw connections between mathematics and athletic achievements over time.
Such problems are often intriguing because they allow us to predict and compare data across different eras, enhancing our understanding of how sports have evolved. It's a reminder that math can help illuminate human progress, even in physical endeavors like the Olympics.

Inequalities in Algebra

Inequalities are a fundamental concept in algebra which allows us to see ranges where certain conditions hold true. In the context of this exercise, we have an inequality that compares the mens' Olympic pole vaulting heights with a given benchmark from the women's event.
Here, the inequality is formulated as follows:

• Given men's height function: \( H(x) = \frac{1}{48} x - 35.83 \)
• We need to solve \( H(x) \leq 5 \) for \( x \).

This involves simple algebraic steps, first by adding 35.83 to both sides to simplify, followed by multiplying to eliminate fractions. Such steps are crucial for determining the years when women’s achievements would have matched men’s results.
Understanding how to solve inequalities can help us in situations where we need a range rather than specific solutions. This is very practical, as real-world problems often demand such flexibility and range rather than fixed outcomes.

Mathematical Modeling

Mathematical modeling is the process of using mathematics to represent, analyze, and interpret real-world scenarios. In this exercise, we use a linear model \( H(x) = \frac{1}{48} x - 35.83 \) to estimate the men's Olympic pole vault heights.
Models like these are essential tools because they allow us to examine and project historical and future trends. They are concise, using variables and functions to summarize complex data. Such models are not limited to athletics but are widespread in other areas such as economics, engineering, and environmental sciences.
The key to effective mathematical modeling is understanding the assumptions and limitations of the model. For instance, our model works well for the specified years but might not hold true outside of this range. This exercise introduces students to the powerful capability of math to represent reality, encouraging them to think critically about how models relate to real-world observations.