Problem 35 - Algebra 1: Concepts and Skills

Question

Use the following information. The air intake b (in liters per minute) of a cyclist on a racing bike can be modeled by \(b=6.37(1.11)^{s},\) where s is the speed of the bike (in miles per hour). Use a calculator to find the cyclist’s air intake if the racing bike is traveling 7 miles per hour, 19 miles per hour, or 25 miles per hour.

Step-by-Step Solution

Answer

The cyclist's air intake would be approximately \(10.4\) litres per minute at 7 mph, \(37.0\) litres per minute at 19 mph, and \(89.2\) litres per minute at 25 mph.

Step 1: Calculate air intake for speed 7 mph

Substitute \(s=7\) into the mathematical model: \(b=6.37(1.11)^{7}\). Use a calculator to find the result.

Step 2: Calculate air intake for speed 19 mph

Substitute \(s=19\) into the mathematical model: \(b=6.37(1.11)^{19}\). Use a calculator to find the result.

Step 3: Calculate air intake for speed 25 mph

Substitute \(s=25\) into the mathematical model: \(b=6.37(1.11)^{25}\). Use a calculator to find the result.

Key Concepts

Mathematical Modeling

Mathematical modeling is like creating a miniature world where we use math to describe how things work in real life. In the context of our exercise, we use a mathematical model to predict the air intake of a cyclist riding a racing bike. The formula given, \[ b = 6.37(1.11)^s \] takes the cyclist's speed \( s \) (in miles per hour) and uses it to determine the air intake \( b \) (in liters per minute). This model helps us understand how different speeds can affect the amount of air a cyclist intakes while cycling. Mathematical models are essential because they allow us to simulate and predict behaviors without direct experimentation. They translate real-world phenomena into mathematical language that we can explore, manipulate, and analyze.

• Provides a way to predict outcomes based on input variables.
• Uses equations to represent real-world scenarios.
• Helps in making informed decisions based on predicted outcomes.

Understanding the structure of the model is key. In this case, 6.37 is a constant factor, and 1.11 is the base of the exponential function influenced by the variable \( s \). Knowing what each part represents can deepen your understanding of the model's role.

Calculator Use in Mathematics

Using a calculator effectively is crucial for working with numbers and mathematical expressions, especially when dealing with exponential functions. In our exercise, the calculator helps execute the power operation efficiently. To compute the result for each cyclist speed given, we follow these steps:

• Input the base of the power: 1.11
• Use the power function (usually denoted as 'x^y' or '^') to raise the base 1.11 to the power of the given speed \( s \).
• Multiply the resulting value by 6.37 to get the air intake \( b \).

A calculator makes this process quick and less error-prone compared to manual computation. When using a calculator, keep in mind to input values accurately and double-check that you've used the correct mathematical operations. Practicing these steps strengthens your calculator skills, making you proficient in handling more complex mathematical problems.

Algebraic Substitution

Algebraic substitution allows us to replace a variable with a specific numerical value to evaluate expressions. In this exercise, we substitute different speeds \( s \) into the equation to find the cyclist's air intake. For example, you replace \( s \) with 7, 19, and 25 for each calculation case:

• Substitute \( s = 7 \) into \( b = 6.37(1.11)^7 \)
• Substitute \( s = 19 \) into \( b = 6.37(1.11)^{19} \)
• Substitute \( s = 25 \) into \( b = 6.37(1.11)^{25} \)

Substitution is a straightforward yet powerful technique in algebra. It helps transform abstract equations into something numerical that can be calculated. By practicing substitution, you improve your ability to work with algebraic expressions, enabling you to solve various mathematical problems more efficiently.