Candle Length Equation: Modeling Burn Time

Have you ever wondered how to predict the length of a burning candle after a certain amount of time? This article will walk you through the process of creating an equation to model this scenario. We'll take a practical example and break it down step by step, so you can understand the logic and apply it to other similar situations.

Understanding the Scenario

Let's imagine we have a candle that is initially 9 inches long. Nelson lights this candle and diligently records its length, denoted as y inches, at different time intervals, measured in x hours. Our goal is to devise an equation that accurately represents the relationship between the number of hours the candle burns (x) and its remaining height (y).

To accomplish this, we'll be given a table of values. This table will show us the candle's length at various points in time. By analyzing these data points, we can discern a pattern and formulate an equation that captures the candle's burning behavior. Before we dive into the actual equation-building, let's first look at the importance of understanding linear equations, slopes, and y-intercepts, as these concepts are fundamental to solving this problem.

Linear Equations: The Foundation

At its core, modeling the candle's burn often boils down to understanding linear equations. Linear equations represent a straight-line relationship between two variables. In our case, these variables are the number of hours the candle burns (x) and the candle's height (y). The beauty of a linear equation is its simplicity – it changes at a constant rate, which makes it predictable. The general form of a linear equation is:

y = mx + b

Where:

• y is the dependent variable (the candle's height)
• x is the independent variable (the time in hours)
• m is the slope (the rate at which the candle burns)
• b is the y-intercept (the initial height of the candle)

Decoding the Slope: The Rate of Change

The slope (m) is the heart of our equation. It tells us how much the candle's height changes for every hour it burns. In mathematical terms, the slope is the "rise over run," or the change in y divided by the change in x. A negative slope indicates that the candle's height is decreasing as time increases, which is exactly what we'd expect. To find the slope, we'll need at least two points from our table of values. These points will give us the 'before' and 'after' snapshots we need to calculate the rate of change.

The Y-intercept: The Starting Point

The y-intercept (b) is our starting point. It's the candle's height when we first light it, or when x (time) is zero. This is often the easiest part to identify because it's usually given directly in the problem or can be easily inferred from the data. In our example, the initial length of the candle (9 inches) is our y-intercept. It’s the value of y when x is 0, marking the beginning of our candle's burning journey.

Analyzing the Table of Values

Let's assume we have the following table of values representing Nelson's observations:

Now, let's dissect this data to build our equation. The table shows a clear trend: as the number of hours increases, the height of the candle decreases. This confirms our suspicion of a negative slope.

Calculating the Slope

To calculate the slope (m), we need to pick two points from the table. Let's choose the points (0, 9) and (1, 7.5). These points represent the candle's height at the beginning and after one hour.

The formula for slope is:

m = (y2 - y1) / (x2 - x1)

Plugging in our values:

m = (7.5 - 9) / (1 - 0) m = -1.5 / 1 m = -1.5

So, our slope is -1.5. This means the candle burns 1.5 inches per hour.

Identifying the Y-intercept

The y-intercept (b) is the height of the candle when x = 0. Looking at the table, we can see that when the candle has burned for 0 hours, its height is 9 inches. Therefore, our y-intercept is 9.

Building the Equation

Now that we have the slope (m = -1.5) and the y-intercept (b = 9), we can plug these values into the slope-intercept form of a linear equation:

y = mx + b

y = -1.5x + 9

This is our equation! It models the length of the candle (y) after burning for x hours.

Testing the Equation

To ensure our equation is accurate, let's test it with a value from the table. For example, let's see if it correctly predicts the candle's height after 3 hours.

Using x = 3:

y = -1.5(3) + 9 y = -4.5 + 9 y = 4.5

Our equation predicts that after 3 hours, the candle will be 4.5 inches tall. This matches the value in our table, giving us confidence in our equation.

Using the Equation for Predictions

Now that we have our equation, we can use it to predict the candle's height at any time. For instance, if we want to know the candle's height after 5 hours:

y = -1.5(5) + 9 y = -7.5 + 9 y = 1.5

According to our model, the candle will be 1.5 inches tall after 5 hours.

Key Takeaways

Modeling real-world scenarios with equations can seem daunting, but by breaking it down into smaller steps, it becomes manageable. Here's a recap of the key steps we followed:

1. Understand the Scenario: Grasp the context and identify the variables involved.
2. Analyze the Data: Look for patterns and relationships in the table of values.
3. Calculate the Slope: Determine the rate of change using two points from the table.
4. Identify the Y-intercept: Find the starting point (the value of y when x = 0).
5. Build the Equation: Plug the slope and y-intercept into the slope-intercept form (y = mx + b).
6. Test the Equation: Verify its accuracy using values from the table.
7. Use for Predictions: Apply the equation to predict future outcomes.

By following these steps, you can confidently model various real-world situations using linear equations. This approach is not just limited to candle burning; it can be applied to anything that exhibits a linear relationship, such as the depreciation of a car or the growth of a plant.

Conclusion

Creating an equation to model the length of a burning candle is a great way to see how math can be used in everyday life. By understanding the concepts of slope and y-intercept, and by carefully analyzing data, we can build equations that accurately represent real-world situations. This skill is invaluable not only in mathematics but also in various fields that require data analysis and prediction.

Remember, the key is to break down the problem into smaller, manageable steps. With practice, you'll become more comfortable with modeling and predicting outcomes in various scenarios.

For further exploration of linear equations and their applications, you might find helpful resources on websites like Khan Academy's Algebra I section.