Problem 37 - Algebra 2

Question

Find the perimeter of a regular pentagon whose sides measure \((2 \sqrt{3}+3 \sqrt{12})\) feet.

Step-by-Step Solution

Answer

The perimeter is \(40 \sqrt{3}\) feet.

Step 1: Simplify the Expression for the Side Length

First, simplify the given expression for the side length, which is \(2 \sqrt{3} + 3 \sqrt{12}\). Note that \(\sqrt{12}\) can be simplified as \(\sqrt{4 \times 3} = 2 \sqrt{3}\). Therefore, the expression becomes:\[2 \sqrt{3} + 3 \times 2 \sqrt{3} = 2 \sqrt{3} + 6 \sqrt{3} = 8 \sqrt{3}.\]

Step 2: Calculate the Perimeter of the Pentagon

A regular pentagon has 5 equal sides. Given the side length is \(8 \sqrt{3}\) feet, the perimeter \(P\) is calculated by multiplying the side length by 5:\[P = 5 \times 8 \sqrt{3} = 40 \sqrt{3}.\]

Key Concepts

Perimeter

When you hear the word "perimeter," think of walking all the way around a shape. It's a way to measure the total length of the boundary around a figure. For any polygon, including regular and irregular shapes, the perimeter is simply the sum of the lengths of all its sides.
A crucial point is that units matter! The perimeter will have the same measurement unit as the side lengths, whether feet, meters, or any other unit.
To calculate the perimeter easily:

• First, determine if the shape is regular (all sides are equal) or irregular (different side lengths).
• For a regular shape, multiply the side length by the number of sides.
• For an irregular shape, add up all the side lengths individually.

Remember, the perimeter gives you the total distance around the figure, making it useful for tasks like fencing a garden or framing a picture.

Pentagon

A pentagon is a five-sided polygon, derived from "penta," meaning five. In geometry, understanding different types of polygons is essential. A regular pentagon has equal side lengths and equal interior angles.
These properties simplify calculations, as you only need one measurement to find the perimeter. Each interior angle of a regular pentagon measures 108 degrees, adding up to a total interior angle sum of 540 degrees for the whole figure.
Here's why pentagons are interesting:

• The shape is often used in architecture and design to create unique structures due to its symmetry and aesthetic appeal.
• Stars and other decorative motifs are commonly based on pentagonal shapes.

Geometric shapes like pentagons are fundamental to both mathematical theory and practical applications.

Simplifying Radical Expressions

When dealing with radical expressions, simplification is often required to make further calculations easier. Simplification involves decreasing the complexity of the expression while keeping its value the same.
Here’s a step-by-step guide to simplify:

• Find prime factors of the number under the radical. This helps determine any perfect squares.
• Rewrite the radical expression using these factors, pulling out any perfect squares.
• Combine and simplify like terms if necessary.

In the given problem, - We started with the radical expression of side length \(2 \sqrt{3} + 3 \sqrt{12}\).- By simplifying \(\sqrt{12}\) to \(2 \sqrt{3}\), we focused on similar radicals.- Finally, this resulted in \(8 \sqrt{3}\), a much simplified form.
This process is crucial because it makes multiplying or adding radicals more straightforward, leading to an accurate result for operations like finding the perimeter.