Calculating the Distance Between Two Poles Using Basic Algebra

Often, we encounter scenarios where we need to calculate the distance between two points, especially when dealing with poles or markers. This article will guide you through a step-by-step approach to calculate the distance between two poles using basic algebra, without the need for trigonometry. Let's break down the problem and solve it.

Understanding the Problem

Given an image or description of two poles, we want to calculate the distance between them. The simplest and most accurate way to do this involves setting up a coordinate system and using the Pythagorean theorem. However, in the absence of an image, we’ll use hypothetical lengths to illustrate the process.

Basic Setup and Assumptions

Let’s assume we have two poles, labeled q and r, with certain lengths λ (longer length) and σ (shorter length). The goal is to find the distance between these poles. For simplicity, we’ll assume the poles are placed on a Cartesian plane with q at the origin (0,0).

Coordinate Geometry and Basic Equations

In the plane, point q is at (0,0). Point r is at some coordinate (αλ, θ), where α is a ratio, and θ is a fixed distance. Point s is at (1-ασ, θ). Point o (which we need to find) is somewhere in the plane, and we will use the Pythagorean theorem to find its coordinates.

Setting Up the Equations

To solve for the distance, we can use the Pythagorean theorem in two different triangles:

Triangle qrs, where the distance is σ. Triangle qps, where the distance is λ.

For triangle qrs, we have:

[ text{Distance} A sqrt{frac{theta^2}{alpha^2}}text{ and }sigma^2 A^2 frac{theta^2}{alpha^2} ]

For triangle qps, we have:

[ text{Distance} A sqrt{frac{theta^2}{(1-alpha)^2}}text{ and }lambda^2 A^2 frac{theta^2}{(1-alpha)^2} ]

Solving the Equations

By setting the two equations for (A^2) equal to each other, we get:

[frac{theta^2}{alpha^2} - frac{theta^2}{(1-alpha)^2} lambda^2 - sigma^2 ]

Further simplification leads to:

[alpha^4 - 2alpha^3 - 2left(frac{theta^2}{lambda^2} - sigma^2right)alpha - frac{theta^2}{lambda^2} sigma^2 0]

This is a quartic equation, and we can solve it using numerical methods or approximation techniques. Given the constraints, we can solve for (alpha) and then find (A).

Example: If (lambda 30), (sigma 20), and (theta 8), we get:

[alpha^4 - 2alpha^3 - 2left(frac{64}{900} - 1right)alpha - frac{64}{900} 1 0 ]

After solving, (alpha approx 0.683), and the distance (A approx 16.212).

Conclusion

This method provides a general approach to calculate the distance between two poles using basic algebra. It requires setting up and solving a quartic equation, which can be approximated or solved numerically. This approach can be applied to various scenarios, provided the lengths and angles are known or can be measured or estimated.

References

Pythagorean Theorem

Quartic Equation