Problem 8 - Calculus with Concepts in Calculus

Question

Determine the distance between the given points. \((\sqrt{6}, \sqrt{3})\) and \((3 \sqrt{6},-\sqrt{3})\)

Step-by-Step Solution

Answer

The distance between the points is 6.

Step 1: Recall the Distance Formula

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\].

Step 2: Plug Points into Formula

Substitute the given points \((\sqrt{6}, \sqrt{3})\) and \((3\sqrt{6}, -\sqrt{3})\) into the distance formula. Here, \(x_1 = \sqrt{6},\ y_1 = \sqrt{3}\), \(x_2 = 3\sqrt{6}, \ y_2 = -\sqrt{3}\).

Step 3: Apply Coordinates to Formula

Calculate each component separately: 1. \(x_2 - x_1 = 3\sqrt{6} - \sqrt{6} = 2\sqrt{6}\).2. \(y_2 - y_1 = -\sqrt{3} - \sqrt{3} = -2\sqrt{3}\).

Step 4: Compute Each Term

Square the results from the previous step: 1. \((x_2 - x_1)^2 = (2\sqrt{6})^2 = 4 \times 6 = 24\).2. \((y_2 - y_1)^2 = (-2\sqrt{3})^2 = 4 \times 3 = 12\).

Step 5: Calculate the Sum

Add the squared terms: \(24 + 12 = 36\).

Step 6: Finalize the Distance

Take the square root of the sum: \(\sqrt{36} = 6\). So, the distance between the points is 6.

Key Concepts

Coordinate Geometry

Coordinate geometry is an integral part of mathematics, providing a bridge between algebra and geometry through the use of coordinates. In this system:

• Each point in the plane is defined by an ordered pair of numbers, usually denoted as \((x, y)\).
• These coordinates identify a specific location on a grid.

Understanding the concept of coordinate geometry allows us to visually interpret equations and solve geometric problems by translating them into algebraic expressions. This geometric framework is key in various fields, including computer graphics, physics simulations, and real-world navigation systems. When working with coordinate geometry, always visualize the plane as a set of horizontal (x-values) and vertical (y-values) lines intersecting to form a grid.

Distance Between Points

Finding the distance between two points on a coordinate plane involves calculating the straight line that connects them. This is where the Distance Formula comes into play:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

• The formula is derived from the Pythagorean theorem.
• It calculates the hypotenuse of a right triangle whose legs run parallel to the axes.

In our specific example, the formula is applied to points \((\sqrt{6}, \sqrt{3})\) and \((3\sqrt{6}, -\sqrt{3})\), illustrating its utility in handling real numbers and square roots efficiently. Mastery of this formula allows you to determine the precise distance in practical applications like mapping the shortest path in navigation software or finding the shortest route in urban planning.

Mathematical Calculation

The process of calculating the distance between two points involves several steps, each requiring careful attention to detail. First, you plug the points' coordinates into the distance formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]For our points

• \(x_1 = \sqrt{6}\), \(y_1 = \sqrt{3}\)
• \(x_2 = 3\sqrt{6}\), \(y_2 = -\sqrt{3}\)

After substituting, calculate the differences:

• \(x_2 - x_1 = 2\sqrt{6}\)
• \(y_2 - y_1 = -2\sqrt{3}\)

Then square each result:

• \((2\sqrt{6})^2 = 24\)
• \((-2\sqrt{3})^2 = 12\)

Add these squares and find their square root to obtain the final distance: \[d = \sqrt{36} = 6\].Calculations like these showcase the beauty of mathematics, where complex real numbers and operations simplify into meaningful results.