Problem 110 - College Algebra

Question

A vertical pole is supported by three wires. Each wire is 13 yards long and is anchored 5 yards from the base of he pole. How far up the pole will the wires be attached?

Step-by-Step Solution

Answer

The wires will be attached approximately \(12\) yards up the pole.

Step 1: Understand the Problem

Visualize the problem as a right triangle, where the pole is the vertical leg, the distance from the base of the pole to where the wires are anchored is the horizontal leg, and the wire is the hypotenuse.

Step 2: Apply the Pythagorean Theorem

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Here, the hypotenuse (c) is 13 yards, and the horizontal leg (a) is 5 yards. We need to find the length of the vertical leg (b). So, \[b = \sqrt{c^2 - a^2}\]

Step 3: Substitute and Solve

Substitute \(c = 13\) yards and \(a = 5\) yards into the equation to obtain \(b = \sqrt{13^2 - 5^2}\). Simplify this to find the value of b.

Step 4: Interpret the Result

The result obtained in Step 3 gives us the length of the vertical side of the triangle, which is the height up the pole where the wires will be attached.

Key Concepts

Understanding Right Triangles

A right triangle is one of the most fundamental geometric shapes. It has three sides, like all triangles, but what makes it unique is that one of these angles is a right angle, measuring 90 degrees. This distinctive angle makes the right triangle a crucial figure in various mathematical applications, especially when using the Pythagorean Theorem.

In a right triangle:

• The side opposite the right angle is the longest and is known as the hypotenuse.
• The other two sides are referred to as the legs. These are typically of different lengths and together they form the right angle.

Understanding the configuration of a right triangle is essential when solving real-world problems, like determining lengths and heights in everyday objects.

The Role of the Hypotenuse

The hypotenuse is often the star of the show in right triangles. It is always opposite the right angle and is the longest side of the triangle. In the context of the exercise, the hypotenuse is represented by the wire that supports the vertical pole, measuring 13 yards.

The importance of the hypotenuse lies in its role in the Pythagorean Theorem, which mathematically establishes a relationship between all three sides of a right triangle. The theorem asserts that:

• The square of the hypotenuse length is equal to the sum of the squares of the other two sides.

This equation forms the basis for finding the length of an unknown side when the lengths of the other two sides are known.

Explaining the Vertical Pole

A vertical pole in this scenario serves as one of the legs of the right triangle. Imagine it as the upright side that forms a 90-degree angle with the ground. This setup mirrors the conditions given in the exercise.

From there:

• The height of the pole (the distance above the ground where the wire attaches) is the key value determined by the exercise.
• By applying the Pythagorean Theorem, you can solve for this vertical side's length using the known lengths of the hypotenuse (wire) and the horizontal leg.

Having the pole as one side of the triangle accurately models this problem, making the concept of right triangles applicable and very practical for real-life solutions.

Calculating the Triangle Sides

Calculating the sides of a right triangle requires understanding the relationship between them as defined by the Pythagorean Theorem. Given the side lengths:

• The hypotenuse is 13 yards (the wire).
• The horizontal side (distance from the base of the pole to where the wire is anchored) is 5 yards.
• The unknown side is the vertical height up the pole where the wire attaches.

To find the length of the unknown side, rearrange the theorem to solve for this missing leg. This means using the formula:

\[ b = \sqrt{c^2 - a^2} \]
Substitute the given values into this formula. Upon calculation, the resulting length gives you the exact height at which the wires are attached to the pole, thus solving the practical problem at hand.