Problem 67 - Intermediate Algebra

Question

Use the data in each table to find an equation that mathematically describes the relationship between the two quantities. $$ \begin{array}{|c|c|} \hline \text { Tower height (ft) } & \text { Height of base (ft) } \\ \hline 15.5 & 5.5 \\ \hline 22 & 12 \\ \hline 25.25 & 15.25 \\ \hline 45.125 & 35.125 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The equation is \( y = x + 10 \).

Step 1: Understand the problem

Look at the given table that shows the tower height and the height of the base in feet. We need to find an equation describing the relationship between these two quantities.

Step 2: Identify the Relationship

Examine how the tower height and base height are related in the table. Notice that the tower height increases linearly as the base height increases, suggesting a linear relationship of the form \(y = mx + b\), where \(y\) is the tower height and \(x\) is the base height.

Step 3: Calculate the Slope

The slope \(m\) is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Take any two pairs of points, for example, (5.5, 15.5) and (12, 22). The slope is \(m = \frac{22 - 15.5}{12 - 5.5} = \frac{6.5}{6.5} = 1\).

Step 4: Derive the Equation

With the slope \(m = 1\), use one of the points to find the y-intercept. Using (5.5, 15.5), substitute into \(y = mx + b\): \(15.5 = 1 \times 5.5 + b\). Solving for \(b\) gives \(b = 10\). Therefore, the equation is \(y = x + 10\).

Step 5: Verify the Equation

Check the derived equation \(y = x + 10\) with other data points. Substitute \(x = 12\), we get \(y = 12 + 10 = 22\), which matches the table. Similarly, all other points satisfy this equation.

Key Concepts

Understanding Slope Calculation

The slope is a vital concept in linear equations. It's like the road map that tells us how steep or flat a line is. In mathematical terms, the slope, represented by \(m\), is the rate at which one quantity changes compared to another. To find the slope of a line, we use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This formula measures the difference in the \(y\)-values over the difference in the \(x\)-values between any two points on the line.

Let's break this down with a practical example. When looking at two points, like \((5.5, 15.5)\) and \((12, 22)\), you subtract the \(y\)-values, giving you \(22 - 15.5\), and the \(x\)-values, giving you \(12 - 5.5\). This leads to the slope calculation \(m = \frac{6.5}{6.5} = 1\). A slope of 1 means that for every unit increase in \(x\) (in this case, the base height), there's a corresponding unit increase in \(y\) (the tower height).

The slope provides insight into the direct relationship between two variables: the tower height and base height here. For data that consistently increases in this manner, the slope confirms our linear relationship suspicions.

Finding the Y-Intercept

The y-intercept is like the starting point of a line on a graph. It's where the line crosses the y-axis. In the context of the equation of a straight line \(y = mx + b\), \(b\) represents the y-intercept.

Determining the y-intercept involves using a known point and the calculated slope. Take the point \((5.5, 15.5)\) on our line. We've already figured out the slope \(m = 1\). By substituting these into the linear equation, we substitute \(x = 5.5\) and \(y = 15.5\):

• \(15.5 = 1 \times 5.5 + b\)
• This simplifies to \(15.5 = 5.5 + b\)
• Solving for \(b\), we find \(b = 10\)

This calculation tells us that when the base height is zero (if we could theoretically reduce it to zero), the tower height would start at 10 feet. This value helps us to plot the starting point of our line on a graph and ensures consistency across other data points.

Interpreting The Linear Relationship

A linear relationship means that the relationship between two variables can be represented by a straight line when plotted on a graph. This is a fundamental concept in algebra and graphing, and it allows us to predict values and understand trends.

In our case, the linear relationship is described by the equation \(y = x + 10\). This relationship tells us that:

• The tower height (\(y\)) increases by 1 foot for every additional foot in base height (\(x\)).
• The base of any tower, when reduced to zero height, would still mean the tower starts at 10 feet, thanks to our y-intercept.

This kind of relationship reveals that changes between these two variables are constant. Linear relationships provide a powerful tool for understanding how two variables interact in a predictable way.