Problem 21 - College Algebra with Modeling and Visualization

Question

Find the exact distance between the two points. Where appropriate, also give approximate results to the nearest hundredth. $$ \left(\frac{1}{2},-\frac{1}{2}\right),\left(\frac{3}{4}, \frac{1}{2}\right) $$

Step-by-Step Solution

Answer

The exact distance is \( \frac{\sqrt{17}}{4} \approx 1.03 \).

Step 1: Identify the Coordinates of the Points

We have two points given: \( \left(\frac{1}{2}, -\frac{1}{2}\right) \) and \( \left(\frac{3}{4}, \frac{1}{2}\right) \). Let's denote the first point as \( (x_1, y_1) \) and the second point as \( (x_2, y_2) \).So, \( x_1 = \frac{1}{2} \), \( y_1 = -\frac{1}{2} \), \( x_2 = \frac{3}{4} \), and \( y_2 = \frac{1}{2} \).

Step 2: Use the Distance Formula

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute the values of \( x_1, y_1, x_2, \) and \( y_2 \) into this formula.

Step 3: Calculate Differences in Coordinates

Calculate the difference between the x-coordinates and y-coordinates:\( x_2 - x_1 = \frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4} \)\( y_2 - y_1 = \frac{1}{2} - (-\frac{1}{2}) = \frac{1}{2} + \frac{1}{2} = 1 \)

Step 4: Compute the Squares of the Differences

Compute the squares of the differences calculated:\((x_2 - x_1)^2 = \left(\frac{1}{4}\right)^2 = \frac{1}{16} \)\((y_2 - y_1)^2 = 1^2 = 1 \)

Step 5: Add the Squares and Take the Square Root

Add the squared differences and take the square root to find the distance:\[d = \sqrt{\frac{1}{16} + 1} = \sqrt{\frac{1}{16} + \frac{16}{16}} = \sqrt{\frac{17}{16}}\]\[d = \sqrt{\frac{17}{16}} = \frac{\sqrt{17}}{4}\]

Step 6: Approximate the Distance

Calculate the approximate value of \( \frac{\sqrt{17}}{4} \). We know \( \sqrt{17} \approx 4.123 \), so:\[d \approx \frac{4.123}{4} = 1.03075 \]Thus, rounding to the nearest hundredth, \( d \approx 1.03 \).

Key Concepts

Coordinate Geometry

Coordinate geometry, often known as analytic geometry, is all about using algebraic formulas to solve geometric problems. By using a coordinate plane, we can describe geometric shapes with equations, which allows us to compute various properties, such as distance. In most problems, you'll encounter points that lie on a two-dimensional grid. Each point is represented by its coordinates, which are an ordered pair of numbers,

• x-coordinate: Measures horizontal distance from a vertical reference line (the Y-axis).
• y-coordinate: Measures vertical distance from a horizontal reference line (the X-axis).

In the exercise at hand, the points \((\frac{1}{2}, -\frac{1}{2})\) and \((\frac{3}{4}, \frac{1}{2})\) are plotted on such a grid.
The horizontal axis (x-axis) and vertical axis (y-axis) divide the plane into four quadrants. These coordinates are crucial in applying the distance formula to find the distance between two points in this geometrical environment.

Exact and Approximate Values

When calculating the distance between two points, it's important to know both exact and approximate values. The exact value arises when we compute the distance with perfect precision, usually involving square root calculations that aren't entirely neat numbers. In this exercise, after substituting into the distance formula, the exact value we obtain is \(\frac{\sqrt{17}}{4}\).However, mathematics often requires approximate values as well, especially when dealing with roots or irrational numbers. Approximate values provide a decimal representation that conveys a real-world measurement. In our case,

• Exact Value: \(\frac{\sqrt{17}}{4}\) is the precise answer derived from the calculation.
• Approximate Value: Since \(\sqrt{17} \approx 4.123\), the approximate distance is \(1.03\) when rounded to the nearest hundredth.

Having both types of values helps us appreciate mathematical purity while maintaining practical use.

Calculating Differences in Coordinates

The first step in using the distance formula is to calculate the differences between the x-coordinates and y-coordinates of the points involved. This is crucial because these differences provide the base measurements we use to determine how far apart the points are horizontally and vertically.
For this exercise, the differences were computed as follows:

• X-coordinates: \(x_2 - x_1 = \frac{3}{4} - \frac{1}{2} = \frac{1}{4}\)
• Y-coordinates: \(y_2 - y_1 = \frac{1}{2} - (-\frac{1}{2}) = 1\)

These differences are used in the following steps where they are squared and then summed up. Thus, calculating the difference is step 1 of creating a basis for finding the exact distance through the distance formula. Once these differences are known, squaring and summing them comes next, followed by taking the square root to find the distance in a neat operation that simplifies applying the distance formula.