Problem 59 - Algebra A Combined Function

Question

Solve. Give exact answers and two-decimal-place approximations where appropriate. A wire is needed to support a vertical pole 15 feet tall. The cable will be anchored to a stake 8 feet from the base of the pole. How much cable is needed?

Step-by-Step Solution

Answer

The wire needed is 17 feet.

Step 1: Understanding the Problem

A vertical pole is 15 feet tall, and a wire needs to support it. The wire is anchored 8 feet from the pole's base. We need to find the wire's length, which forms a right triangle with the pole and the ground.

Step 2: Setting Up the Equation

We are dealing with a right triangle where the pole is the vertical leg (15 feet), the distance from the base to the anchor stake is the horizontal leg (8 feet), and the wire is the hypotenuse. We use the Pythagorean Theorem: \[c^2 = a^2 + b^2\]where \(a\) = 15 feet, \(b\) = 8 feet, and \(c\) is the length of the wire.

Step 3: Calculating the Wire Length

Plug in the values: \[c^2 = 15^2 + 8^2\]Simplify and solve for \(c\): \[c^2 = 225 + 64 = 289\]\[c = \sqrt{289} = 17\]The exact length of the wire needed is 17 feet.

Step 4: Final Answer

The wire needed is 17 feet long, which is both the exact answer and the two-decimal-place approximation.

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of its angles measures exactly 90 degrees. This type of triangle is significant in various mathematical problems, particularly those involving the Pythagorean Theorem. In a right triangle, there are three sides: the base, the height (often referred to as the perpendicular side), and the hypotenuse. The hypotenuse is always opposite the right angle and is the longest side of the triangle.

In our problem, the right triangle is formed by the vertical pole, the horizontal ground, and the supporting wire (which is the hypotenuse). Understanding the properties of a right triangle helps us solve the exercise by applying suitable geometric and algebraic principles.

Geometry Problem

Geometry involves understanding shapes, their properties, and spatial relationships. In this exercise, the task is to determine the length of a wire needed to support a pole using geometric principles.

This involves:

• Recognizing a right triangle scenario.
• Identifying the known sides (the pole and the distance to the stake).
• Applying the Pythagorean Theorem to find the unknown side (the wire's length).

Solving geometry problems often requires logical thinking and visualization of the scenario to set up appropriate equations, as we did here using the context of a real-world setup.

Hypotenuse Calculation

The process of finding the hypotenuse involves applying the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (\(c^2\)) is equal to the sum of the squares of the other two sides (\(a^2 + b^2\)).

For this problem:

• The vertical pole is one leg (\(a = 15\) feet).
• The ground distance to the anchor point is the other leg (\(b = 8\) feet).
• The hypotenuse (\(c\)) is the wire's length.

Using the formula, \[c^2 = 15^2 + 8^2\]we calculate\[c^2 = 225 + 64 = 289\]Taking the square root gives us\[c = \sqrt{289} = 17\]Thus, 17 feet of wire is needed.

Algebraic Solution

The algebraic approach to solving this problem involves setting up and solving equations based on the known properties of the right triangle. Starting with the Pythagorean Theorem, identify the given measures and the one unknown.

Steps involved:

• Write down the equation \(c^2 = a^2 + b^2\) with known values for \(a\) and \(b\).
• Substitute: \(c^2 = 225 + 64\).
• Simplify, to find \(c^2 = 289\).
• Find \(c = \sqrt{289}\) to get \(c = 17\).

Thus, through basic algebraic manipulation, we deduced the wire length is 17 feet.

Mathematical Reasoning

Mathematical reasoning involves logical thinking to understand and solve problems. In this scenario, it revolves around understanding the relationship between the triangle's sides and applying the Pythagorean Theorem logically.

Key reasoning points included:

• Recognizing the problem's geometric nature and identifying where to apply algebra.
• Understanding that the hypotenuse in this real-life application (wire) is the longest triangle side and solving accordingly.
• Translating a practical setup into mathematical expressions and solving using known theorems.

Through these processes, we used deductive reasoning to determine the exact measurement needed for the supporting wire.