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Chapter 1: Problem 2

Pole vault: The height of the winning pole vault in the early years of the modern Olympic Games can be modeled as a function of time by the formula \(H=0.05 t+3.3\) Here \(t\) is the number of years since 1900 , and \(H\) is the winning height in meters. (One meter is \(39.37\) inches.) a. Calculate \(H(4)\) and explain in practical terms what your answer means. b. By how much did the height of the winning pole vault increase from 1900 to 1904 ? From 1904 to 1908 ?

Short Answer

Expert verified
The winning height in 1904 was 3.5 meters. From 1900 to 1904, the height increased by 0.2 meters, and from 1904 to 1908, it also increased by 0.2 meters.

Step by step solution

01

Understand the Formula

The formula given is \(H = 0.05t + 3.3\), where \(H\) represents the height in meters and \(t\) is the number of years since 1900. The task is to calculate the height for a given number of years since 1900 and understand the increase in height over certain periods.
02

Calculate H(4)

To calculate \(H(4)\), substitute \(t = 4\) into the formula. That gives us \(H = 0.05 \times 4 + 3.3\). Calculate the equivalent: \(H = 0.2 + 3.3 = 3.5\) meters.
03

Interpret H(4) Result

The result, \(H(4) = 3.5\) meters, means that 4 years after 1900, in the year 1904, the winning height of the pole vault was 3.5 meters.
04

Calculate Height Increase from 1900 to 1904

From 1900 to 1904, we need the height increase. \(H(0)\) is the height in 1900. Substitute \(t = 0\) into \(H = 0.05t + 3.3\) to get \(H(0) = 3.3\) meters. We already found \(H(4) = 3.5\) meters for 1904. The increase is \(3.5 - 3.3 = 0.2\) meters.
05

Calculate Height Increase from 1904 to 1908

Substitute \(t = 8\) into the formula: \(H(8) = 0.05 \times 8 + 3.3\). Calculate it to get \(H(8) = 0.4 + 3.3 = 3.7\) meters. The height in 1904 was \(H(4) = 3.5\) meters. The increase is \(3.7 - 3.5 = 0.2\) meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Linear Functions
Linear functions are like a straight line when you draw them on a graph. They show a constant rate of change between two variables. In the case of our pole vault problem, the formula \(H = 0.05t + 3.3\) is a linear function. The variable \(t\) represents time in years since 1900, and \(H\) stands for height in meters. The key characteristics of linear functions include:
  • A constant slope, or rate of change. Here, it's \(0.05\) means for every year that passes, the height increases by \(0.05\) meters.
  • An initial value or y-intercept. In this formula, \(3.3\) meters is the starting height at the year 1900.
Understanding these parts of the equation helps explain how the height of pole vaults changes over time. Linear functions make such calculations easy and predictable.
Mathematical Modeling in Sports
Mathematical modeling is a powerful tool that uses formulas and functions to represent the real world in a simplified way. In sports, like the pole vault, mathematical models can help us understand trends and predict future outcomes. Let's see how it applies in our example:
  • The model \(H = 0.05t + 3.3\) is built to estimate pole vault heights across specific years.
  • It simplifies a complex world, considering only time and height, making it easier to analyze trends.
  • This model helps predict increases in height, allowing athletes and coaches to set realistic goals.
By stripping away distractions and focusing on key variables, mathematical modeling enables focused analysis and predictions in sporting events.
Algebra Applications in Real-Life Scenarios
Algebra isn't just for math class; it's used in real-life scenarios all the time. The pole vault problem is a great way to see how algebra applications work in reality.

In this scenario, we use algebra to:
  • Calculate future performance - like using \(H(4)\) to predict the 1904 pole vault height.
  • Determine progression - seeing the increase between 1900 to 1904 and further, allows us to measure improvement.
Working out these kinds of problems with algebra can also boost critical thinking skills. You'll learn how to break problems down, apply logical steps, and find meaningful solutions. Even outside of sports, you can use algebra to make calculations in finance, construction, and technology.

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Most popular questions from this chapter

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