Problem 65 - College Algebra with Corequisite Support

Question

A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating \(\sqrt{90,000+160,000}\). What is the length of the guy wire?

Step-by-Step Solution

Answer

The length of the guy wire is 500 feet.

Step 1: Understand the Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. For this problem: \(a^2 + b^2 = c^2\), where \(a\) is the base (90,000 feet²), \(b\) is the height (160,000 feet²), and \(c\) is the hypotenuse (the guy wire).

Step 2: Set up the equation for the guy wire

We combine the given squares of the legs of the triangle to find the square of the length of the hypotenuse. The equation becomes \(c^2 = 90,000 + 160,000\).

Step 3: Simplify the equation

Add the two given squares together: \(90,000 + 160,000 = 250,000\). Therefore, \(c^2 = 250,000\).

Step 4: Solve for the length of the guy wire

Find \(c\) by taking the square root of 250,000. This gives \(c = \sqrt{250,000}\).

Step 5: Calculate the square root

Compute \(\sqrt{250,000}\) to find the actual length of the guy wire. Using a calculator, \(\sqrt{250,000} = 500\).

Step 6: Final answer

The length of the guy wire is 500 feet.

Key Concepts

Understanding Right-Angled Triangles

A right-angled triangle is a type of triangle that has one angle exactly equal to 90 degrees. This 90-degree angle makes it unique and fundamental to many areas of mathematics, especially geometry. In a right-angled triangle, there are three sides - the base, the height, and the hypotenuse.
Right-angled triangles are essential for solving various real-world problems. They appear in architecture, engineering, and even in simple tasks like measuring heights. The notable feature of these triangles is their practical use in calculations due to the Pythagorean Theorem.
Understanding these triangles can simplify complex problems by allowing us to break down real-world objects into simpler, calculable shapes.

• They consist of three sides: two shorter sides connected by the right angle, and a longer side known as the hypotenuse.
• They are commonly solved in geometry using the Pythagorean Theorem.
• The right angle is often marked with a small square in diagrams.
  

The balance between practical application and mathematical calculation makes right-angled triangles an invaluable tool in your learning toolkit.

The Role of the Hypotenuse

The hypotenuse is the longest side of a right-angled triangle, always located opposite the right angle. In simple terms, when you stand in front of a right angle, the hypotenuse stretches away - think of it as the diagonal in a rectangle made by the triangle.
The hypotenuse is crucial because it represents the result of the Pythagorean Theorem. For our original problem, the hypotenuse is the guy wire, calculated using the theorem's formula \(a^2 + b^2 = c^2\). Where \c\ symbolizes the hypotenuse.

• This longest side helps us link and calculate unknown sides of triangles using given measurements.
• In practical scenarios, it signifies the direct route or distance between two points, making it a handy calculation tool.
• In the described exercise, discovering the hypotenuse reveals the guy wire length necessary for the structure's stability.
  

Knowing how to identify and use the hypotenuse can provide direct solutions to many geometric problems, as it serves as the key measurement in right-angled triangles.

Basic Concepts of Geometry

Geometry is a branch of mathematics revolving around the shape, size, and relative position of figures and the properties of space. It's the study of how shapes fit together and interact in the plane or three-dimensional space around us.
By using geometry, we can predict the shape and configuration of everything from simple lines to complex structures like bridges and buildings. In the exercise, geometry provided a clearer view for pinpointing precise measurements of the guy wire.

• It involves various shapes and measures, including triangles, circles, and polygons.
• Mathematical calculations like the Pythagorean Theorem aid in identifying distances and angles within figures.
• Understanding geometric concepts helps solve real-world engineering and architectural challenges.
  

When you grasp the fundamentals of geometry, you gain the capability to view and analyze the world in a structured way. This understanding aids not only in academic pursuits but also in everyday problem-solving and professional tasks.