Height of a Flagpole A flagpole is secured on opposite sides...

Height of a Flagpole A flagpole is secured on opposite sides by two guy wires, each of which is $5 \mathrm{ft}$ longer than the pole. The distance between the points where the wires are fixed to the ground is equal to the length of one guy wire. How tall is the flagpole (to the nearest inch)?

Key Concepts

Algebraic Manipulation

Algebraic manipulation is the technique used to simplify and solve equations derived from mathematical models. It involves operations like expanding, factoring, and rearranging terms to isolate unknown variables. Mastery of algebraic manipulation is critical for making progress from the initial problem setup to the final solution.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in Euclidean geometry that relates the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem provides the basis for solving problems involving distances and heights in right-angled contexts.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation that often arises when setting up mathematical models for physical problems. In many geometric applications, including those involving right triangles, the relationships between the sides lead to equations that are quadratic in form. Solving these equations is essential with methods such as factoring or using the quadratic formula.

Problem Modeling in Geometry

Problem modeling in geometry involves converting a real-world situation into a mathematical framework by identifying relevant geometric relationships. This process includes determining the shapes involved—such as right triangles in this case—and setting up equations that accurately reflect the physical dimensions and constraints of the scenario.

Transcript

00:01 We have a flagpole with a height which i've called h, and it's held up by two wires, each of which has a length 5 feet longer than the flagpole, which i've called h plus 5.

00:09 Additionally, the wires touch the ground h plus 5 units away from each other, with the flagpole right in the middle.

00:15 Now let's solve for h.

00:16 You may notice here that we can break this up into two different right triangles.

00:20 So if you look at the left one, you'll notice that it has sides equal to h, h plus 5, and this third side, which is half of h plus 5, or h plus 5.

00:30 Plus 5 over 2.

00:32 So since this is a right triangle we can use the pythagorean theorem.

00:37 So h squared plus h plus 5 over 2 squared is equal to h plus 5 squared.

00:49 That is a squared plus b squared equals c squared.

00:53 Now let's rearrange this equation because it's a bit foreboding.

00:56 So we have h squared plus now we can square the top and the bottom separately.

01:01 The bottom, the squared, that's easy, that's just four.

01:04 The top we need to use the distributive property, which you may have learned is the foil method.

01:08 This comes out to h squared plus 10h plus 25.

01:13 And this is equal to, once again, distributing h squared plus 10h plus 25.

01:22 Okay, now there's an h squared on both sides...