Problem 4 - Calculus: One and Several Variables

Question

Find the distance between the points. $$P_{0}(2,7), \quad P_{1}(-4,7)$$

Step-by-Step Solution

Answer

The distance between the points P_{0}(2,7) and P_{1}(-4,7) is 6 units.

Step 1: Identify the coordinates

The coordinates of points P_{0} and P_{1} are given. For P_{0}, 2 is the x-coordinate and 7 is the y-coordinate. Similarly, for P_{1}, -4 is the x-coordinate and 7 is the y-coordinate.

Step 2: Substitute in the formula

Now we will substitute these coordinates into the distance formula. In our case, x_1=2, y_1=7, x_2=-4, y_2=7. So, the distance d becomes \sqrt{(-4-2)^2 + (7-7)^2}.

Step 3: Simplify

Simplifying the equation gives us, d= \sqrt{(-6)^2 + 0} = \sqrt{36}.

Step 4: Find the square root

Finding the square root gives us the answer, d = 6 units.

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of mathematics where we use a set of values, known as coordinates, to represent positions on a plane. This method helps us easily calculate distances and understand the spatial relationship between geometric figures. Each point on a plane is represented by a pair of numbers called coordinates:

• The first number is the x-coordinate, which shows the horizontal position.
• The second number is the y-coordinate, which shows the vertical position.

For example, in the problem given, we have two points,

• \(P_0(2, 7)\): This point has an x-coordinate of 2 and a y-coordinate of 7.
• \(P_1(-4, 7)\): This point has an x-coordinate of -4 and the same y-coordinate of 7.

This means both points lie on a line parallel to the horizontal (x-axis), making it easy to find the distance between them by comparing their x-coordinates.

Distance Calculation

The distance between two points in coordinate geometry can be calculated using the Distance Formula. This formula is derived from the Pythagorean Theorem and gives us a direct way to assess how far apart two points are. The Distance Formula is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Let's understand how this works with the points from our example:

• The coordinates for \(P_0\) are \((x_1, y_1)\) where \(x_1 = 2\), \(y_1 = 7\).
• The coordinates for \(P_1\) are \((x_2, y_2)\) where \(x_2 = -4\), \(y_2 = 7\).

Substituting these values into the formula, we consider the change in the x-values:

• \((x_2 - x_1) = (-4 - 2) = -6\)
• \((y_2 - y_1) = (7 - 7) = 0\)

Now, plug these differences into the formula: \[d = \sqrt{(-6)^2 + 0^2} = \sqrt{36}\]Thus, we find that the distance \(d\) is 6 units.

Square Root

In mathematics, a square root is a value that, when multiplied by itself, gives the original number. This concept is crucial for finding the actual distance between points using the Distance Formula. Once we calculate the squared differences in coordinates, we take the square root to finalize the measurement of distance.
For example, in the problem we solved, after we calculated the squared difference as 36:

• We find the square root of 36, written as \( \sqrt{36} \).
• The square root of 36 is 6, because \(6 \times 6 = 36\).

Therefore, the final step in finding the distance is to compute this square root, ensuring we interpret the formula correctly and provide the true distance on a plane. Understanding square roots not only helps in distance calculations but also in various other areas of mathematics, such as solving quadratic equations and simplifying expressions.