Problem 15 - 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. $$ (7,-4),(9,1) $$

Step-by-Step Solution

Answer

The exact distance is \(\sqrt{29}\), which is approximately 5.39.

Step 1: Identify Coordinates

Identify the coordinates of the two given points. The coordinates are Point 1: \((7, -4)\) and Point 2: \((9, 1)\).

Step 2: Use the Distance Formula

The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substitute the coordinates into this formula: \(d = \sqrt{(9 - 7)^2 + (1 + 4)^2}\).

Step 3: Simplify the Expression

Simplify the expression inside the square root: \(d = \sqrt{2^2 + 5^2}\). Calculate \(2^2 = 4\) and \(5^2 = 25\).

Step 4: Calculate the Sum

Add the squared values: \(4 + 25 = 29\). So, \(d = \sqrt{29}\).

Step 5: Approximate the Result

Calculate the approximate value of \(\sqrt{29}\) to the nearest hundredth. \(\sqrt{29} \approx 5.39\).

Key Concepts

Coordinate Geometry: Connecting Points and Distances

In coordinate geometry, we use a system called the Cartesian coordinate system to locate points on a plane using two numbers, often known as x and y coordinates. This system helps us visualize geometric concepts and solve problems related to distance and direction.
For example, in the given task, we have two points:

• Point 1: (7, -4)
• Point 2: (9, 1)

These coordinates represent the horizontal (x) and vertical (y) positions of each point. Using these coordinates, we can calculate the distance between points by applying a critical tool in coordinate geometry: the distance formula. This shifts the abstract concept of distance into a concrete mathematical calculation.

Square Root Calculation: Simplifying Distances

The calculation of the square root is essential when determining distance between points in coordinate geometry. The distance formula involves taking a square root of the sum of squared differences in coordinates.
The distance formula, \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\], allows us to find the distance between two points on a plane. In our exercise, by substituting the given points’ coordinates, \[d = \sqrt{(9 - 7)^2 + (1 + 4)^2} = \sqrt{4 + 25} = \sqrt{29}\].
Learning to easily compute square roots involves understanding both their basic definition and how they apply in geometry. Knowing how to handle the square root symbol can significantly ease calculations, as not every root comes out a whole number. Calculators and estimation skills are often used to simplify these square roots, connecting us to our next concept.

Approximation in Mathematics: From Exactness to Practical Use

When calculating distances, we often need to round our results to a practical level of precision, especially with irrational numbers like the square root of 29. Approximations make complex calculations manageable and provide a level of precision that's useful in real-life applications, such as engineering and architecture.
In our task, once reaching \[\sqrt{29}\], we used approximation to the nearest hundredth: \[\sqrt{29} \approx 5.39\]. Rounding is a common mathematical practice used to make results more comprehensible and practical.
Key aspects of approximation include:

• Understanding decimal places and significant figures.
• Knowing how to correctly round up or down.
• Recognizing which level of precision is necessary for the task.

These approximations allow for greater ease and clarity in communicating mathematical results.