Problem 59 - Algebra and Trigonometry

Question

Clint is building a wooden swing set for his children. Each supporting end of the swing set is to be an A-frame constructed with two 10 -foot-long 4 by 4 's joined at a \(45^{\circ}\) angle. To prevent the swing set from tipping over, Clint wants to secure the base of each A-frame to concrete footings. How far apart should the footings for each A-frame be?

Step-by-Step Solution

Answer

The footings should be \(10 \sqrt{2} \) feet apart.

Step 1: Identify the Components

Clint is creating an A-frame for the swing set with two 10-foot-long pieces of wood. The pieces are joined at a 45-degree angle on each side.

Step 2: Visualize the A-frame Structure

Imagine the A-frame as two sides of an isosceles triangle, with the peak at the top. The angle between the two pieces at the top is 90 degrees (since two 45-degree angles make 90 degrees when joined).

Step 3: Break Down the Triangle

Each side of the A-frame is 10 feet long, and we need to find the distance between the two footings, which is the base of the triangle. This triangle can be split into two right triangles, each with a hypotenuse of 10 feet and an angle of 45 degrees.

Step 4: Use Trigonometry

In the right triangle, the relationship between the hypotenuse (10 feet) and the legs can be found using trigonometric functions. For a 45-degree right triangle, the legs are equal, and each leg is given by \( \frac{10}{\sqrt{2}} \).

Step 5: Calculate the Base

Each leg of the right triangles represents half of the base of the isosceles triangle. Therefore, the base is twice one leg: \( 2 \cdot \frac{10}{\sqrt{2}} = 2 \cdot 5 \sqrt{2} = 10 \sqrt{2} \) feet.

Key Concepts

A-frame construction

When building structures like a swing set, the A-frame design is a popular choice. This type of construction uses two long beams, joined together at the top to form a shape similar to the letter 'A'.
In our problem, Clint's swing set uses two 10-foot-long wooden beams joined at the top at an angle of 45 degrees to ensure stability.
This setup not only provides a sturdy structure but also a defined geometric shape that can be analyzed using simple trigonometry.

isosceles triangle

The A-frame can be visualized as an isosceles triangle, where the two sides are of equal length. In Clint's case, both wooden beams serve as the equal sides, each being 10 feet long.
The peak of this triangle, where the beams meet, has an angle of 90 degrees, as it is formed by joining two 45-degree angles. Understanding this is crucial as it helps to simplify the overall problem into manageable geometric parts.

right triangle trigonometry

To solve for the base of the isosceles triangle (distance between the footings), we can split it into two right triangles.
Each right triangle has:
- A hypotenuse of 10 feet
- An angle of 45 degrees at the base

Right triangle trigonometry tells us that in a 45-degree triangle, the legs are equal. Using this property, we use trigonometric functions to determine the lengths of these legs. For a hypotenuse of 10 feet, each leg will be: \( \frac{10}{\frac{\text{legs}}{\text{hypotenuse}}} = \frac{10}{\frac{\text{leg}}{\texthypotenuse}} = base \)

triangle angle relationships

Knowing how angles interact within a triangle helps to solve problems involving triangles. In Clint's A-frame, the joining of two 45-degree angles forms a 90-degree angle at the top.
Understanding these angle relationships is fundamental as they define the type and properties of the triangles we need to solve. This is essential for accurate calculations and constructing stable structures.

concrete footings

Lastly, securing the A-frame to the ground with concrete footings prevents the swing set from tipping over.
Concrete footings act as solid bases for the structure and they spread the load evenly to the ground. In our scenario, the distance between these footings, which forms the base of the triangle, ensures stability.
Through our calculations, we found that this distance should be \(2 \times 5 \times \frac{10}{\text{legs}} \sqrt2 \). Placing the footings at this distance will ensure the swing set remains upright and stable.