Problem 64 - Elementary Algebra

Question

Set up an algebraic equation and use it to solve. An 18-foot ladder leaning against a building reaches a height of 17 feet. How far is the base of the ladder from the wall? Round to the nearest tenth of a foot.

Step-by-Step Solution

Answer

The base of the ladder is approximately 5.9 feet from the wall.

Step 1: Understand the Problem

We have a right triangle formed by the ladder, the wall, and the ground. The ladder is the hypotenuse, the height reached by the ladder is one leg, and the distance from the wall is the other leg.

Step 2: Identify Known Values

The length of the ladder (hypotenuse) is 18 feet, and the height the ladder reaches on the wall (one leg) is 17 feet.

Step 3: Use the Pythagorean Theorem

The Pythagorean theorem states that for a right triangle, \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the legs. Here, \(a = 17\) feet and \(c = 18\) feet.

Step 4: Set Up the Equation

Let \(b\) be the distance from the wall. The equation becomes \(17^2 + b^2 = 18^2\).

Step 5: Simplify the Equation

Calculate \(17^2 = 289\) and \(18^2 = 324\). Substitute into the equation: \(289 + b^2 = 324\).

Step 6: Solve for \(b^2\)

Subtract 289 from 324 to find \(b^2\): \(b^2 = 324 - 289 = 35\).

Step 7: Find \(b\)

Take the square root of both sides to solve for \(b\): \(b = \sqrt{35}\).

Step 8: Calculate and Round

Calculate \(\sqrt{35} \approx 5.916\). Round to the nearest tenth: \(b \approx 5.9\).

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This angle, known as the right angle, divides the triangle into two smaller angles. The three sides of a right triangle consist of:

• The two legs: These are the sides that form the right angle.
• The hypotenuse: This is the longest side, located opposite the right angle.

Understanding right triangles is critical because they frequently appear in problems involving distance and height, like those with ladders leaning against walls. Recognizing the right angle helps identify which side is the hypotenuse and which are the legs. This knowledge is essential before applying the Pythagorean theorem.

Hypotenuse

In a right triangle, the hypotenuse is the side opposite the right angle and is always the longest side. When solving problems using the Pythagorean theorem, identifying the hypotenuse is crucial.
For example, in the problem with the ladder:

• The ladder itself forms the hypotenuse at 18 feet.

Knowing the hypotenuse allows us to apply the Pythagorean theorem correctly, aiding in solving for unknown side lengths. Remember that the hypotenuse cannot be shorter than the other sides, as it must stretch across the triangle's largest span.

Algebraic Equations

Algebraic equations form the backbone of solving problems involving right triangles and the Pythagorean theorem. An algebraic equation expresses relationships using symbols and mathematical operations.
In the context of a right triangle, the Pythagorean theorem gives us the equation:
\[a^2 + b^2 = c^2 \]where:

• \(a\) and \(b\) represent the legs.
• \(c\) represents the hypotenuse.

The process:

• Identify known values for some sides.
• Substitute them into the equation.
• Solve for the unknown side.

In the example with the ladder, we set \(17^2 + b^2 = 18^2\), which simplifies to \(b^2 = 35\). Solving \(b = \sqrt{35}\) gives the distance from the wall. Algebraic equations help translate geometric concepts into solvable numeric problems.