Problem 101 - Elementary and Intermediate Algebra

Question

The function \(H(s)=-47.73+107.38 \ln s\) approximates the heart rate (in beats/minute) for an Olympic-class cross country skier traveling at \(s\) miles per hour, where \(s>5\) mph. Find the heart rate of a skier traveling at a rate of 7.5 miles per hour. (Source: btc.ontana.edu/Olympics/physiology)

Step-by-Step Solution

Answer

The heart rate of the skier is approximately 169 beats per minute.

Step 1: Identify Given Values

Identify the given value for the speed of the skier. In this problem, we are given the speed \( s = 7.5 \) mph.

Step 2: Substitute into the Function

Substitute \( s = 7.5 \) into the function \( H(s) = -47.73 + 107.38 \ln s \). This yields: \( H(7.5) = -47.73 + 107.38 \ln(7.5) \).

Step 3: Evaluate the Natural Logarithm

Calculate the natural logarithm of 7.5 using a calculator: \( \ln(7.5) \approx 2.0149 \).

Step 4: Calculate the Heart Rate

Substitute \( \ln(7.5) \approx 2.0149 \) into the equation: \( H(7.5) = -47.73 + 107.38 \times 2.0149 \). This results in: \(H(7.5) = -47.73 + 216.409 \).

Step 5: Simplify to Find the Heart Rate

Perform the arithmetic: \( H(7.5) = 216.409 - 47.73 = 168.679 \).

Step 6: Round the Final Answer

Round the calculated heart rate to the nearest whole number. The skier's heart rate is approximately 169 beats per minute.

Key Concepts

Introduction to Functions

In mathematics, a *function* is a name given to a process or a rule that assigns each element in a given set, known as the domain, to one and only one element in another set, called the range. The beauty of functions lies in their ability to describe relationships between varying quantities. Functions can be linear, quadratic, or involve more complex operations such as logarithms and exponentials. Each function can be expressed mathematically, enabling us to predict outputs based on different inputs.
For instance, in the case of the function provided:

• The function is represented as: \( H(s) = -47.73 + 107.38 \ln s \), where \( s \) is the speed of the skier.
• Here, \( s \) is the input or the independent variable, and \( H(s) \) is the output or the dependent variable. The function describes how the heart rate of a skier is influenced by their speed.

Functions like these are central to building models that describe real-world phenomena, allowing us to make predictions and understand how one quantity affects another.

Understanding Natural Logarithms

Natural logarithms are a specific type of logarithm with a base known as "e" (approximately 2.71828). The natural logarithm of a number is a powerful mathematical tool used to scale down large numbers and to solve equations involving exponential growth or decay.
When we see \( \ln s \) in the function \( H(s) = -47.73 + 107.38 \ln s \), it signifies the application of natural logarithms on the speed variable \( s \). Natural logarithms help make the relationship between speed and heart rate more manageable by transforming multiplicative relationships into additive ones, which are easier to handle.

• Using a calculator, we find that \( \ln(7.5) \approx 2.0149 \), a critical part of computing the heart rate function.
• This transformation provides insights into small changes in speed and their effects on heart rate, especially in high-performance sports scenarios like skiing.

Mathematical Modeling of Real-world Phenomena

Mathematical modeling is a means of representing real-world systems using mathematical equations. By translating physical, biological, or social phenomena into mathematical terms, we can analyze and simulate their behavior, gaining insights without needing physical trials.
In the problem of modeling a skier's heart rate, we see how the function \( H(s) = -47.73 + 107.38 \ln s \) encapsulates the relation between speed and physiological response.

• The modeling highlights how changes in speed affect heart rate, allowing trainers and athletes to optimize performance.
• Such relationships are derived from empirical data, where observations are synthesized into a coherent function that describes the dynamic between input and output variables.
• This process illustrates the reliance on mathematical modeling in sports science and other fields to simulate conditions and predict outcomes, making strategic planning and execution feasible.

Ultimately, mathematical modeling is a bridge connecting theoretical mathematics with practical applications, emphasizing clarity in complex systems.