Problem 29 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. A building is 2 feet from a 9 -foot fence that surrounds the property. A worker wants to wash a window in the building 13 feet from the ground. He plans to place a ladder over the fence so that it rests against the building. He decides he should place the ladder at least 8 feet from the fence for stability. To the nearest foot, how long a ladder will he need?

Step-by-Step Solution

Answer

The ladder needs to be approximately 16 feet long.

Step 1: Understanding the Problem

We need to find the length of a ladder that can rest against the building, taking into account a fence that is 2 feet away and 9 feet tall. The ladder must also be 8 feet from the fence for stability while reaching a height of 13 feet to the window.

Step 2: Visualizing the Setup

Draw a right triangle where one side is the distance from the fence to the building (2 feet), plus the distance the ladder is placed from the fence (8 feet), making a total horizontal distance of 10 feet. The vertical side is the height from the ground to the window (13 feet). The ladder acts as the hypotenuse of this triangle.

Step 3: Applying the Pythagorean Theorem

Use the Pythagorean theorem, which is expressed as \( a^2 + b^2 = c^2 \), where \( a \) is the horizontal distance (10 feet), \( b \) is the vertical distance (13 feet), and \( c \) is the hypotenuse of the triangle (the ladder's length).

Step 4: Calculate the Ladder Length

Substitute the values into the Pythagorean theorem: \( (10)^2 + (13)^2 = c^2 \). Calculate \( 10^2 = 100 \), and \( 13^2 = 169 \), then add these results: \( 100 + 169 = 269 \). Finally, solve for \( c \) by taking the square root: \( c = \sqrt{269} \).

Step 5: Approximate the Length of the Ladder

Calculate \( \sqrt{269} \), which is approximately 16.4. Round this value to the nearest foot to get the length of the ladder required.

Key Concepts

Understanding Right Triangles

Right triangles are a special type of triangle where one of the angles measures exactly 90 degrees. This characteristic allows us to apply the Pythagorean theorem, a fundamental principle in geometry. In a right triangle, the side opposite the right angle is called the hypotenuse. The other two sides are known as the legs. The Pythagorean theorem states that the square of the hypotenuse (\[ c \] ) is equal to the sum of the squares of the other two sides (\[ a \] and \[ b \] ).
For example, if we have a right triangle with legs measuring 10 feet and 13 feet, we can find the hypotenuse by calculating:

• \( a^2 + b^2 = c^2 \)
• \( 10^2 + 13^2 = c^2 \)
• \( 100 + 169 = c^2 \)
• \( 269 = c^2 \)

Taking the square root gives us the length of the hypotenuse, which is crucial for solving real-world problems like the ladder scenario.

Solving the Ladder Problem

The ladder problem is a classic example in geometry where practical concerns meet mathematical principles. In this scenario, we need to figure out how long a ladder should be to reach a certain height while overcoming obstacles like a fence. Understanding the setup is crucial: imagine placing a ladder against a building that is 13 feet high at the point of contact, with a 9-foot fence standing 2 feet away from the building.
The ladder must rest 8 feet from the fence for safety, resulting in a total horizontal distance of 10 feet (2 feet to the fence and 8 feet from the fence to the ladder's base). This setup forms a right triangle with:

• One leg as the horizontal distance (10 feet)
• The other leg as the vertical distance to the window (13 feet)
• The ladder itself being the hypotenuse

By applying the Pythagorean theorem, we can derive the required ladder length.

Identifying and Solving Geometry Problems

Geometry problems often involve visualizing and setting up a scenario accurately. These problems can range from simple shapes to complex three-dimensional objects. In the context of the ladder problem, visualization helps us understand how the ladder, the building, and the fence fit together.
To solve such problems:

• Start by identifying all elements of the geometry, like measurements and angles.
• Draw a detailed diagram to visualize relationships between different components.
• Apply geometric principles such as the Pythagorean theorem to find missing measures.

This systematic approach helps clarify complex relationships and leads to accurate solutions, as seen in the ladder problem where we have calculated the ladder's required length by setting up a right triangle.

Calculating the Hypotenuse

The hypotenuse is the longest side of a right triangle, opposite the right angle. Calculating it is a key aspect of many geometry challenges, such as determining the correct length of a ladder as seen in our exercise.Using the Pythagorean theorem:

• Determine the lengths of the other two sides (legs) of the triangle, in this case, 10 feet and 13 feet.
• Apply the formula: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.
• Compute: \( 10^2 + 13^2 = c^2 \)
• Sum up the squares: \( 100 + 169 = 269 \)
• Find the square root: \( c = \sqrt{269} \approx 16.4 \)

Finally, round to the nearest whole number, giving us a ladder length of about 16 feet. This calculation demonstrates how the hypotenuse finds practical application in real-world scenarios such as the ladder problem.