Problem 63 - College Algebra with Corequisite Support

Question

A man on the top of a building wants to have a guy wire extend to a point on the ground \(20 \mathrm{ft}\) from the building. To the nearest foot, how long will the wire have to be if the building is \(50 \mathrm{ft}\) tall?

Step-by-Step Solution

Answer

The guy wire should be approximately 54 feet long.

Step 1: Understand the Problem

Visualize the scenario as a right triangle where the building is the vertical leg, the distance from the building to the point on the ground is the horizontal leg, and the guy wire is the hypotenuse. The lengths are given: the building (vertical leg) is 50 ft tall, and the distance from the building (horizontal leg) is 20 ft.

Step 2: Apply the Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle, the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)). Here, \(a = 50\) ft, \(b = 20\) ft, and \(c\) is the length of the wire. So, apply the formula: \(c^2 = a^2 + b^2\).

Step 3: Substitute Known Values

Substitute the known values into the Pythagorean Theorem: \(c^2 = 50^2 + 20^2\). Calculating, we obtain \(c^2 = 2500 + 400\).

Step 4: Simplify and Solve for c

Add the results from the substitution: \(c^2 = 2900\). Take the square root of both sides to solve for \(c\): \(c = \sqrt{2900}\).

Step 5: Round to the Nearest Foot

Calculate the square root \(\sqrt{2900} \approx 53.85\). Round to the nearest foot, resulting in \(54\) ft.

Key Concepts

Right Triangle

When we talk about a right triangle, it's all about a specific kind of triangle that has one 90-degree angle. This is called the right angle. The other two angles in a right triangle are acute, meaning they are less than 90 degrees each.
In a right triangle, the side opposite the right angle is known as the hypotenuse. This is always the longest side of the triangle. The other two sides are called the legs. One leg is positioned vertically, often representing height, while the other stretches horizontally, often representing the base or distance.

• In our example, the vertical leg is the building itself, which is 50 feet tall.
• The horizontal leg is the distance from the building to the point on the ground, which is 20 feet.
• The hypotenuse is the guy wire extending from the top of the building to the ground.

Understanding the layout of right triangles is crucial, especially when applying mathematical theories like the Pythagorean Theorem.

Geometry

Geometry is the branch of mathematics that concerns itself with the properties and relations of points, lines, surfaces, and solids. It's all about shapes and spaces.
In the context of our problem, we're applying geometric principles to figure out the right length for the wire. This involves understanding and calculating lengths in a two-dimensional space, specifically through triangles.
In our scenario, geometry helps us:

• Visualize the problem: Seeing the building, the distance on the ground, and the wire as a right triangle.
• Apply formulas: Understanding how to use the Pythagorean Theorem to find missing lengths.
• Interpret results: Translating our calculations into real-world measurements, such as feet in our example.

Relying on geometry allows us to solve practical problems by applying theoretical math concepts.

Algebra

Algebra is like the toolbox of mathematics. It allows us to use symbols and equations to represent and solve problems. It’s all about finding unknowns and working with variables.
In our guy wire problem, algebra comes into play through the Pythagorean Theorem. We use symbols to represent the sides of the triangle and solve for the hypotenuse (c).
Let's see how:

• We define our sides: Let the building height be \(a = 50\) ft, and the ground distance \(b = 20\) ft.
• Our goal is to find \(c\), the hypotenuse: We use the formula \(c^2 = a^2 + b^2\).
• Substitute the known values: \(c^2 = 50^2 + 20^2\) leads us to solve \(c = \sqrt{2900}\).

Algebra helps us get to the final number, which is rounded to 54 feet in this problem. It provides a structured approach to solving for unknowns in mathematical scenarios. This makes solving geometric problems not just possible but straightforward.