Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

In exercise physiology, aerobic power \(P\) is defined in terms of maximum oxygen intake. For altitudes up to 1800 meters, aerobic power is optimal-that is, \(100 \%\). Beyond 1800 meters, \(P\) decreases linearly from the maximum of \(100 \%\) to a value near \(40 \%\) at 5000 meters. (a) Express aerobic power \(P\) in terms of altitude \(h\) (in meters) for \(1800 \leq h \leq 5000\). (b) Estimate aerobic power in Mexico City (altitude: 2400 meters), the site of the 1968 Summer Olympic Games.

Step-by-Step Solution

Answer

Aerobic power is 88.75% at 2400 meters.

Step 1: Identify the points for the linear equation

Given that aerobic power decreases linearly from 100% at 1800 meters to 40% at 5000 meters, we can identify two key points on the line: \((1800, 100)\) and \((5000, 40)\). These will help us find the equation of the line.

Step 2: Calculate the slope of the line

The slope \(m\) of the line can be calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Plugging in the points \((1800, 100)\) and \((5000, 40)\), we have: \[ m = \frac{40 - 100}{5000 - 1800} = \frac{-60}{3200} = -\frac{3}{160} \]

Step 3: Write the linear equation in point-slope form

Using the point-slope form of a line \(y - y_1 = m(x - x_1)\), we substitute \((x_1, y_1) = (1800, 100)\) and \(m = -\frac{3}{160}\), so:\[ P - 100 = -\frac{3}{160}(h - 1800) \]

Step 4: Simplify the equation to the slope-intercept form

Distribute the slope in the equation from Step 3:\[ P - 100 = -\frac{3}{160}h + \frac{3}{160} \times 1800 \] Calculate the constant term:\[ P = -\frac{3}{160}h + 33.75 + 100 \]\[ P = -\frac{3}{160}h + 133.75 \]This is the equation for aerobic power \(P\) in terms of altitude \(h\).

Step 5: Substitute the altitude of Mexico City into the equation

To find the aerobic power at Mexico City's altitude (2400 meters), use the equation from Step 4:\[ P = -\frac{3}{160}(2400) + 133.75 \]Calculate \(-\frac{3}{160} \times 2400\):\[ P = -45 + 133.75 \] \[ P = 88.75 \] So, the aerobic power \(P\) at 2400 meters is estimated to be 88.75%.

Key Concepts

Linear Equation

A linear equation is a vital concept in mathematics. It is an equation that makes a straight line when it is graphed on a coordinate plane. In its simplest form, it can be written as: \[ y = mx + c \] Where:

• \(y\) is the dependent variable.
• \(x\) is the independent variable.
• \(m\) represents the slope of the line.
• \(c\) is the y-intercept, the point where the line crosses the y-axis.

Linear equations are particularly useful for modeling relationships that show consistent change, like the decrease in aerobic power with increasing altitude.
They form the backbone for solving real-world problems involving changes over time or space.

Point-Slope Form

The point-slope form is one way to express the equation of a line. It is particularly useful when you know a point on the line and the slope. The formula is:\[ y - y_1 = m(x - x_1) \] Here:

• \((x_1, y_1)\) is a specific point on the line.
• \(m\) is the slope of the line.

This form is incredibly helpful for quickly deriving an equation when dealing with real-world scenarios, such as calculating the decline of aerobic power with altitude changes. By knowing just one point and the overall slope, you can easily write the equation that describes the situation.

Slope-Intercept Form

The slope-intercept form is a common and easy to understand way to express linear equations. It is given by: \[ y = mx + b \] In this form:

• \(m\) is the slope, indicating how steep the line is.
• \(b\) is the y-intercept, where the line crosses the y-axis.

This form is particularly useful when you need to quickly identify both the starting point and the rate of change from a linear function.
For the problem at hand, converting a linear equation to this form enables quick predictions of aerobic power at various altitudes.

Altitude Impact

Altitude can significantly impact physiological processes, including aerobic power, which is a measure of the body's ability to take in and use oxygen. At higher altitudes, there is less oxygen in the air, meaning less oxygen is available to the body. This is why aerobic power often decreases with increasing altitude.

• Below 1800 meters, aerobic power remains optimal.
• Above that altitude, it decreases linearly to about 40% of its optimal value at 5000 meters.

This concept is incredibly important for athletes or anyone operating at higher elevations, as they need to account for this reduced capability in their activities or training programs.

Oxygen Intake Measurement

Measuring oxygen intake is essential for understanding aerobic power. It refers to the maximum quantity of oxygen that the body can utilize during intense activity. This is often measured in environments that test endurance and is a key performance indicator for athletes.

• Oxygen intake typically decreases at higher altitudes.
• This can impact physical performance due to less oxygen reaching muscles.

However, by understanding the relationship between altitude and oxygen intake through equations, individuals can better prepare for and adapt to high-altitude conditions. This is why calculations like the ones demonstrated in the original exercise are valuable tools for athletes, doctors, and scientists alike.