Problem 6 - Single Variable Calculus

Question

Find the length of the line segment joining the two given points by finding the equation of the line and using Equation (2). Then check your answer by using the distance formula. $$ (-1,-2) \text { and }(3,6) $$

Step-by-Step Solution

Answer

The length of the line segment joining the two given points is \(4\sqrt{5}\).

Step 1: Find the slope of the line

To find the equation of a line, we must first find the slope. Given the two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope can be obtained as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the given points \((-1, -2)\) and \((3, 6)\), we can calculate the slope: \[ m = \frac{6 - (-2)}{3 - (-1)} = \frac{8}{4} = 2 \]

Step 2: Find the equation of the line

Now that we have the slope, we can use the point-slope form to find the equation of the line: \[ y - y_1 = m(x - x_1) \] Using the slope, \(m = 2\) and the point \((-1, -2)\), we get: \[ y - (-2) = 2(x - (-1)) \] Simplifying the equation, we get: \[ y + 2 = 2x + 2 \] So the equation of the line is \(y = 2x\).

Step 3: Use Equation (2) to find the length of the line segment

Equation (2) is not provided, so this step cannot be completed. We will proceed to Step 4 to verify using the distance formula.

Step 4: Verify the length using the distance formula

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Using the given points \((-1, -2)\) and \((3, 6)\), we can calculate the distance: \[ d = \sqrt{(3 - (-1))^2 + (6 - (-2))^2} = \sqrt{4^2 + 8^2} = \sqrt{16 + 64} = \sqrt{80} \] Thus, the length of the line segment is \(\sqrt{80}\), which can be simplified as \(4\sqrt{5}\).

Key Concepts

Understanding the Distance Formula

Measuring the length between two points is like drawing an invisible straight line and asking, 'How long is this?' The distance formula does just that in a mathematical way. It's not magic, although it seems like it when it works so perfectly.

Imagine you want to build a bridge between two cliffs, and you stand on one cliff holding a tape measure. There's obviously no way to stretch it across, right? But if you know exactly how far apart the cliffs are on the map (that's your x difference) and how much higher or lower one cliff is (your y difference), then voilà, the distance formula comes to the rescue:
d = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

It's like you take the horizontal and vertical distances, form a right-angled triangle, and solve for the hypotenuse. Bingo, you get the length of your bridge. For our problem, this translates to:
\[\sqrt{(3 - (-1))^2 + (6 - (-2))^2}\] which neatly gives us \(4\sqrt{5}\). So, apply it like a magic trick next time you need a measurement between any two points in space, and you'll look like a mathematical wizard!

The Gentle Slope of a Line

Ever watched a hillside and wondered how steep it is? Or maybe you've taken a slide and felt the thrill as gravity pulls you down the slope. That's what the slope of a line is about—it's a number that tells us how 'steep' a line is.

Imagine you're drawing a line on graph paper from one dot to another. If you switch one square right (along the x-axis), the slope tells you how many squares up or down (along the y-axis) you need to go. It's like rising over run, or in maths language:
m = \(\frac{y_2 - y_1}{x_2 - x_1}\)

For our two points (-1,-2) and (3,6), plug and play into the formula gives us a slope of 2. That means for every step right, we step two units up. It's a decent climb but not too tough—perfect for a short hike or, in our case, to find out how our line leans in the mathematical world.

Point-Slope Form: The Trusty Map

Okay, so you've climbed a hill and you're taking in the view. Now you want to draw a path for others to follow your route. That's where the point-slope form of a line comes in. Think of it as your trusty map, guiding anyone to follow in your exact footsteps.

It starts with a single point—where you're standing—and the slope, like the path's steepness you've just calculated. The formula looks like this:
y - y_1 = m(x - x_1)

Let's say our point is (-1, -2) and we've found our slope to be 2. We plug those into the formula, and it tells us the recipe for the entire line's path:
\[y + 2 = 2(x + 1)\] which simplifies to \(y = 2x\). That's the guide for anyone who wants to walk the same line from anywhere on the 'map' (or graph). This equation is like leaving breadcrumbs or setting up signposts along the way—simple and incredibly helpful for tracing the route!