Problem 54 - Algebra Form and Function

Question

A cyclist's distance in \(\mathrm{km}\) from the finish line, \(t\) minutes after reaching the flat, is given by \(f(t)=45-\) \(0.5(t-12)\) (a) What is the practical meaning of the constants 12 , \(45,\) and \(0.5 ?\) (b) Express \(f\) in a form that clearly shows the distance from the start of the flat to the finish line.

Step-by-Step Solution

Answer

The constant 12 represents the time when the cyclist reaches the flat area, 45 represents the distance in km from the finish line when the cyclist first reaches the flat area, and 0.5 represents the cyclist's speed on the flat terrain in km/minute. b) Express the function in a form that shows the distance from the start of the flat to the finish line. The function can be rewritten as \(f(t) = 51 - 0.5t\), which shows that initially, at the start of the flat terrain (t=12), the cyclist is 51 km from the finish line and moves towards it at a rate of 0.5 km per minute.

Step 1: Understand the Meaning of the Constants

Let's first take a look at the given function: $$f(t) = 45 - 0.5(t-12)$$ There are three constants in the function: 12, 45, and 0.5 - \(12\): This constant shows the time when the cyclist reaches the flat area. This means that t=12 corresponds to the time when the cyclist starts on the flat terrain. - \(45\): This constant represents the distance in km from the finish line when the cyclist first reaches the flat area (at t=12 minutes). - \(0.5\): This constant represents the speed of the cyclist on the flat terrain in km/minute. It indicates that the cyclist is moving toward the finish line at a rate of 0.5 km per minute.

Step 2: Express the Function in a Form that Clearly Shows the Distance from the Start of the Flat to the Finish Line

We want to express the function in a form that clearly shows the distance from the start of the flat. Let's first find the value of the function at t=12: $$f(12) = 45 - 0.5(12-12) = 45$$ At t=12, the distance from the finish line is 45 km. We can now rewrite the function as follows: $$f(t) = 45 - 0.5(t-12) = 45 - 0.5t + 6 = 51 - 0.5t$$ The new expression of the function shows that initially, at the start of the flat terrain (t=12), the cyclist is 51 km from the finish line. From this point, the cyclist starts moving towards the finish line at a rate of 0.5 km per minute.

Key Concepts

Linear Functions

Linear functions are mathematical expressions that describe a straight-line relationship between two variables, typically of the form \(y = mx + b\). In the context of the cyclist distance function, the linear formula is given as:

• \(f(t) = 45 - 0.5(t-12)\)
• Here, \(f(t)\) represents the distance from the finish line in kilometers.
• \(t\) is the time in minutes after reaching the flat.

This linear function helps in predicting the cyclist's distance from the finish line at any given time using a straight line. The "\(m\)" in our typical linear expression is replaced by \(-0.5\) in the cyclist function, representing a constant speed. Linear functions are crucial because they give us a clear, predictable pattern of how variables change together. The term "\(b\)", which is \(45\) in our function, shows where our line intersects the y-axis (distance from finish line at the beginning). Understanding this form allows us to quickly interpret how changes in time affect the cyclist's distance.

Constants in Functions

In any function, constants play a pivotal role in defining specific characteristics of the relationship. In the cyclist's distance function, the constants \(12\), \(45\), and \(0.5\) each have significant meanings:

• **12:** This constant signifies the time when the cyclist reaches the flat terrain. It marks the start point of the function in time.
• **45:** Represents the initial distance from the finish line when the cyclist first hits the flat, measured in kilometers. This is your starting point distance-wise.
• **0.5:** Indicates the cyclist's speed on the flat terrain in km/minute. This speed remains constant, providing a steady prediction of how the cyclist's position changes over time.

Constants are essential as they provide a fixed reference. They help us understand when events begin, how far or close the starting point is, and the rate of movement. These factors allow us to accurately model and predict outcomes.

Distance-Time Relationship

The relationship between distance and time is a fundamental concept in understanding motion, often expressed in terms of functions. For the cyclist:

• The function \(f(t) = 45 - 0.5(t-12)\) shows how distance from the finish line decreases over time.
• A negative rate, \(-0.5\), demonstrates that as time progresses, the distance gets less, meaning the cyclist is getting closer to the finish line.
• The subtractive form of \((t-12)\) reflects that the cyclist only starts this flat journey after 12 minutes.

Visualizing the distance-time relationship helps in creating a clear path of the cyclist's motion. By understanding the slope of the function, \(-0.5\), we can see that it tells us how fast the cyclist is moving relative to time. The steeper the slope (more negative), the faster the cyclist approaches the finish line. Such relationships are instrumental in physics and everyday life, allowing us to solve problems related to speed, duration, and distance efficiently. Using such linear models, students can confidently predict future positions and plan based on constant velocities.