What Is The Rate Of Change Of A Biking Trail That Begins At The Coordinates $(-3,14)$ And Ends At $(6,-1)$?

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In mathematics, determining the rate of change between two points is a fundamental concept with wide-ranging applications. This principle is especially useful in real-world scenarios, such as mapping the slope or gradient of a path. In this article, we will delve into a specific problem involving a biking trail, where we need to calculate the rate of change, also known as the slope, of a straight path given its starting and ending coordinates. This problem is a classic example of applying the slope formula, and understanding how to solve it can provide valuable insights into linear equations and their practical uses. Let's explore how we can effectively determine the rate of change for this biking trail and what the result signifies.

Understanding the Problem: Biking Trail Coordinates and Rate of Change

The problem presents a scenario where a section of a biking trail begins at the coordinates (−3,14)(-3, 14) and follows a straight path that ends at the coordinates (6,−1)(6, -1). The objective is to find the rate of change of this biking trail. In mathematical terms, the rate of change is synonymous with the slope of the line segment connecting these two points. The slope represents the steepness and direction of the line, indicating how much the yy-coordinate changes for each unit change in the xx-coordinate. To solve this problem, we will use the slope formula, which is a cornerstone of coordinate geometry. This formula allows us to calculate the slope (mm) using the coordinates of two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2). Understanding the problem setup is crucial as it lays the foundation for applying the correct formula and interpreting the result in the context of the biking trail.

Applying the Slope Formula: Step-by-Step Calculation

To calculate the rate of change, we utilize the slope formula, which is expressed as:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Where:

  • mm represents the slope or rate of change.
  • (x1,y1)(x_1, y_1) are the coordinates of the starting point.
  • (x2,y2)(x_2, y_2) are the coordinates of the ending point.

In our biking trail problem, the starting point is (−3,14)(-3, 14) and the ending point is (6,−1)(6, -1). Let's assign these values to the variables in the formula:

  • x1=−3x_1 = -3
  • y1=14y_1 = 14
  • x2=6x_2 = 6
  • y2=−1y_2 = -1

Now, substitute these values into the slope formula:

m=−1−146−(−3)m = \frac{-1 - 14}{6 - (-3)}

Simplify the expression:

m=−156+3m = \frac{-15}{6 + 3}

m=−159m = \frac{-15}{9}

Reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

m=−15÷39÷3m = \frac{-15 ÷ 3}{9 ÷ 3}

m=−53m = \frac{-5}{3}

Therefore, the rate of change of the biking trail is −53-\frac{5}{3}. This result indicates that for every 3 units of horizontal change, the trail descends 5 units vertically. The negative sign signifies a downward slope, which means the trail is declining in elevation as you move from the starting point to the ending point. This step-by-step calculation demonstrates the practical application of the slope formula in determining the rate of change between two points.

Interpreting the Result: Rate of Change in the Context of the Biking Trail

The calculated rate of change of −53-\frac{5}{3} provides crucial information about the biking trail's characteristics. The negative sign indicates that the trail slopes downward from the starting point (−3,14)(-3, 14) to the ending point (6,−1)(6, -1). In practical terms, this means that a cyclist traveling along this section of the trail would be descending. The magnitude of the slope, 53\frac{5}{3}, tells us about the steepness of the trail. For every 3 units of horizontal distance covered, the trail drops 5 units vertically. This is a relatively steep decline, which might be challenging for some cyclists. Understanding the rate of change in this context allows us to visualize the trail's profile and anticipate the physical demands it might pose. A steeper negative slope implies a more rapid descent, while a gentler slope would indicate a less drastic change in elevation. Therefore, the rate of change is not just a mathematical value but a meaningful descriptor of the trail's physical characteristics.

Conclusion: Significance of the Rate of Change Calculation

In summary, determining the rate of change of the biking trail is a valuable exercise in applying mathematical concepts to real-world scenarios. By using the slope formula, we found that the rate of change is −53-\frac{5}{3}. This result tells us that the trail has a downward slope, with a vertical drop of 5 units for every 3 units of horizontal distance. This information is crucial for understanding the trail's steepness and the direction of elevation change. The process of calculating the rate of change involves several key steps, including understanding the problem, applying the slope formula, simplifying the expression, and interpreting the result in context. This problem exemplifies how mathematical principles can be used to analyze and describe physical characteristics, such as the slope of a path. Mastering these calculations enhances our ability to interpret and interact with the world around us, whether we are planning a bike ride or analyzing more complex systems.

The final answer is (J) −53-\frac{5}{3}