Problem 62 - Intermediate Algebra

Question

A wire is to be attached to support a telephone pole. Because of surrounding buildings, sidewalks, and roadways, the wire must be anchored exactly 15 feet from the base of the pole. Telephone company workers have only 30 feet of cable, and 2 feet of that must be used to attach the cable to the pole and to the stake on the ground. How high from the base of the pole can the wire be attached?

Step-by-Step Solution

Answer

The wire can be attached approximately 23.63 feet above the base.

Step 1: Understand the Problem

We have a right triangle where the base along the ground is 15 feet, the hypotenuse is the cable of 28 feet after accounting for attachments, and the height (adjacent side) is the unknown variable where the wire attaches to the pole.

Step 2: Set Up the Right Triangle Variables

Let's denote the height from the base of the pole where the wire is attached as \( h \). We have a right triangle with one leg (base) of 15 feet and the hypotenuse (the cable) reduced to 28 feet. Thus, we are solving for \( h \).

Step 3: Use the Pythagorean Theorem

Apply the Pythagorean theorem: \[ a^2 + b^2 = c^2 \]Where \( a = 15 \) feet, \( b = h \), and \( c = 28 \) feet. Substituting the known values gives us:\[ 15^2 + h^2 = 28^2 \]

Step 4: Solve for the Height

Calculate the squares: \[ 15^2 = 225 \text{ and } 28^2 = 784 \]Then plug these into the equation: \[ 225 + h^2 = 784 \]Subtract 225 from both sides to solve for \( h^2 \): \[ h^2 = 784 - 225 = 559 \]Take the square root to find \( h \):\[ h = \sqrt{559} \approx 23.63 \text{ feet} \]

Step 5: Conclusion

Thus, the wire can be attached approximately 23.63 feet above the base of the pole.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle measuring exactly 90 degrees. This unique property makes it a fundamental part of geometry and useful in various practical applications.

In the case of our exercise, imagine a triangle formed by the base of the telephone pole (ground), the height at which the wire is attached, and the wire itself. This forms a right triangle where:

• The base is the distance from the pole to the point where the wire is anchored (15 feet).
• The height is the distance from the base of the pole to where the wire is attached vertically (unknown in the beginning).
• The hypotenuse is the wire itself, less the length used for attachment (28 feet).

Right triangles are crucial because they allow us to use special mathematical principles, like the Pythagorean Theorem, to find unknown side lengths when two sides are known.

Solving Equations

Solving equations is a critical skill in mathematics, especially when dealing with right triangles. When faced with an equation, the goal is to find the value of the unknown variable that makes the equation true.

For our exercise, we use the equation derived from the Pythagorean Theorem: \[ a^2 + b^2 = c^2 \]Given:

• \( a = 15 \) (base of the triangle)
• \( b = h \) (unknown height)
• \( c = 28 \) (hypotenuse)

Plugging in the known values gives the equation:\[ 15^2 + h^2 = 28^2 \]Continuing the solution,

• Compute \( 15^2 = 225 \) and \( 28^2 = 784 \).
• Rearrange to get \( h^2 = 784 - 225 \).
• Simplify to \( h^2 = 559 \).
• Find the square root to get \( h \approx 23.63 \) ft.

Thus, solving equations involves understanding how to manipulate them until you isolate the unknown variable.

Trigonometry

Trigonometry is the branch of mathematics dealing with the relationships between the angles and sides of triangles. Although we don't dive deeply into trigonometric functions here, understanding the basic application in right triangle problems is valuable.

In the context of this exercise, it's less about using sine, cosine, or tangent functions and more about understanding relationships within the right triangle itself. Trigonometry helps us:

• Define the properties of right triangles, essentially making problems like this solvable.
• Use ratios of triangles to find missing length values when combined with the Pythagorean theorem.

While the problem is predominantly solved using the Pythagorean Theorem, the underlying trigonometric principles make the visualization and real-world application clearer. Trigonometry is powerful, providing tools that help solve practical problems related to right triangles, such as supporting structures like telephone poles.