Problem 43 - Algebra 2 and Trigonometry

Question

The area of a square is 150 square feet. Express the length of a side of the square in simplest radical form.

Step-by-Step Solution

Answer

The side of the square is \( 5\sqrt{6} \) feet.

Step 1: Understanding the Problem

We are given the area of a square as 150 square feet. We need to express the length of one side of the square in simplest radical form.

Step 2: Formula for the Area of a Square

The formula to calculate the area of a square is given by: \( A = s^2 \), where \( A \) is the area and \( s \) is the length of a side of the square. We need to find \( s \) in simplest radical form.

Step 3: Set Up the Equation

Since the area \( A \) is 150 square feet, we can set up the equation: \( s^2 = 150 \). To find \( s \), we need to solve this equation for \( s \).

Step 4: Solve for the Side Length

Take the square root of both sides of the equation to solve for \( s \): \( s = \sqrt{150} \).

Step 5: Simplify the Radical

Express \( \sqrt{150} \) in simplest radical form. Start by breaking \( 150 \) into its prime factors: \( 150 = 2 \times 3 \times 5^2 \). The square root becomes: \( \sqrt{150} = \sqrt{2 \times 3 \times 5^2} = 5 \sqrt{6} \).

Step 6: Final Solution

Thus, the length of a side of the square is \( 5\sqrt{6} \) feet in simplest radical form.

Key Concepts

Area of a Square

The area of a square helps us determine how much surface the square covers. To calculate it, we use a simple and elegant formula:

• Formula: The area, denoted as \( A \), is calculated as \( A = s^2 \), where \( s \) represents the length of one side of the square.
• This formula arises because all sides of a square have the same length, and the area measures the total number of square units inside that square.

Knowing the area let's us backtrack to find the side length if we need to, using the concept of the square root.

Simplest Radical Form

Simplifying a square root means expressing it in its simplest radical form. This process involves two main steps: recognizing perfect squares within a number and separating them out.

• Consider a number under the square root, like 150. The goal is to break it down into factors where we can easily identify perfect squares.
• For example, since 150 is not a perfect square by itself, we simplify \( \sqrt{150} \) by finding its prime factors, and then look for pairs.
• We identify \( 5^2 \) as a perfect square within 150, and thus: \( \sqrt{150} = \sqrt{2 \times 3 \times 5^2} = \sqrt{5^2} \cdot \sqrt{6} = 5 \sqrt{6} \).

This form is termed the 'simplest radical form' because it can't be simplified further without losing its exactness.

Prime Factorization

Prime factorization is the process of breaking down a composite number into the product of its prime numbers. This technique is incredibly useful when simplifying square roots.

• Every composite number can be represented uniquely as a product of primes. For instance, let's consider the number 150.
• We systematically divide it by the smallest possible prime numbers: start with 2, dividing yields 75. Next, 75 is divisible by 3, yielding 25. Finally, 25 is divisible by 5, which results in 5. So, the prime factors are 2, 3, and \(5^2\).
• This decomposition helps find perfect squares within a number, aiding in expressing square roots in simpler form.

Understanding this concept is essential for simplifying radicals and is a fundamental skill in arithmetic and algebra.