Problem 90 - Algebra and Trigonometry

Question

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)?

Step-by-Step Solution

Answer

The height of the flagpole is approximately 388 inches.

Step 1: Understand the Problem

We have a flagpole of unknown height, denoted as \(h\). Two guy wires are used to secure the flagpole on opposite sides, such that each wire is 5 feet longer than the flagpole height. Moreover, the distance between the points where these wires are fixed on the ground is equal to the length of one guy wire.

Step 2: Define the Variables

Let's denote the height of the flagpole as \(h\) feet. Consequently, each guy wire is \(h + 5\) feet long. The distance on the ground between the points where the wires are fixed is also \(h + 5\) feet.

Step 3: Apply the Right Triangle Property

The wire, the ground, and the pole form a right triangle. Therefore, according to the Pythagorean theorem: \[(h + 5)^2 = h^2 + \left(\frac{h + 5}{2}\right)^2\] The base of the triangle, where the wires are fixed, is \(\frac{h + 5}{2}\) each for symmetry.

Step 4: Rearrange and Solve the Equation

Simplify the equation: \[(h + 5)^2 = h^2 + \left(\frac{h + 5}{2}\right)^2\] \[= h^2 + \frac{(h + 5)^2}{4}\]Multiply through by 4 to remove the fraction:\[4(h + 5)^2 = 4h^2 + (h + 5)^2\]\[3(h + 5)^2 = 4h^2\]Expand and simplify:\[3(h^2 + 10h + 25) = 4h^2\]\[3h^2 + 30h + 75 = 4h^2\]Rearrange to:\[4h^2 - 3h^2 - 30h - 75 = 0\]\[h^2 - 30h - 75 = 0\]

Step 5: Solve the Quadratic Equation

Use the quadratic formula to solve for \(h\): \[h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a = 1\), \(b = -30\), and \(c = -75\). \[h = \frac{-(-30) \pm \sqrt{(-30)^2 - 4 \cdot 1 \cdot (-75)}}{2 \cdot 1}\]\[h = \frac{30 \pm \sqrt{900 + 300}}{2}\]\[h = \frac{30 \pm \sqrt{1200}}{2}\]\[h = \frac{30 \pm 34.64}{2}\] Take the positive root since height cannot be negative:\[h = \frac{30 + 34.64}{2} \approx 32.32\] feet.

Step 6: Convert Height to Inches

To convert the height to inches, multiply by 12. \[32.32 \text{ ft} \times 12 = 387.84\text{ inches}\] Rounding to the nearest inch gives approximately 388 inches.

Key Concepts

Pythagorean theorem

The Pythagorean Theorem is a fundamental principle used to explore the relationships within right triangles. In this problem, it helps us find the height of a flagpole by examining the configuration of the guy wires, the flagpole, and the ground. According to the theorem, in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Mathematically, it is expressed as:

• For triangle sides: \[c^2 = a^2 + b^2\]
• In our problem: \[(h + 5)^2 = h^2 + \left(\frac{h + 5}{2}\right)^2\]

Here, the hypotenuse is represented by the guy wire, and the other two sides are the flagpole and half the base distance between the wires. This relationship allows us to set up an equation where we can solve for the unknown height of the flagpole.

quadratic equation

Quadratic equations appear when you need to solve for unknowns where the variables are squared. In the height of the flagpole problem, once the Pythagorean theorem is applied, a quadratic equation emerges:

• The quadratic equation derived is:\[h^2 - 30h - 75 = 0\]
• It stems from expanding and simplifying the right triangle's squared expressions and rearranging terms.

To solve a quadratic equation, you can use the quadratic formula:\[h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In our specific example, the coefficients are:

• a = 1, b = -30, c = -75

Plug these values into the formula to solve for the height, ensuring you take the positive root since a negative height doesn't make sense in this context.

right triangles

Right triangles are an essential concept in geometry, especially when dealing with problems involving height and distance. They consist of three sides and three angles, one of which is always a right angle (90 degrees). In this type of triangle:

• The longest side is called the hypotenuse.
• The other two sides are known as the legs.

In the flagpole problem, each configuration of the flagpole with its guy wire forms a right triangle. The wire acts as the hypotenuse, while the flagpole itself and half the ground distance make up the legs. Understanding this arrangement helps in using the Pythagorean theorem and setting up a quadratic equation to find unknown measurements effectively.