Problem 80 - Algebra 1: Concepts and Skills

Question

Find the length of a side \(s\) of a square that has the same area as a rectangle that is 12 centimeters wide and 33 centimeters long. Write your solution in simplest form.

Step-by-Step Solution

Answer

The length of the side of the square is \(6 \sqrt{11} cm\).

Step 1: Calculate the area of the rectangle

The area of a rectangle is determined by multiplying its length and its width. In this case, the length of the rectangle is 33 cm and the width is 12 cm. Hence, the area of the rectangle is \(33 cm \times 12 cm = 396 cm^2 \).

Step 2: Find the side length of the square

The area of the square is equal to the square of the side length. Since the square has the same area as the rectangle (\(396 cm^2\)), we can find the side length by taking the square root of the area. Therefore, \(s = \sqrt{396} cm \).

Step 3: Simplify the side length

To simplify the square root, factorize 396 into prime factors. The prime factorization of 396 is \(2 \times 2 \times 3 \times 3 \times 11\). We can write this as \((2^2) \times (3^2) \times 11\), which simplifies to \( (2 \times 3) \sqrt{11} \). So \(s = 6 \sqrt{11} cm \).

Key Concepts

Area of a Rectangle

To calculate the area of a rectangle, multiply the length of the rectangle by its width. This provides a measure of the total space inside the rectangle.
For our original problem, the rectangle has a length of 33 centimeters and a width of 12 centimeters.
The formula to find the area is:

• The area formula is: \[\text{Area} = \text{Length} \times \text{Width}\]
• Using the values from the problem: \[33 \text{ cm} \times 12 \text{ cm} = 396 \text{ cm}^2\]

The area of 396 square centimeters represents the total amount of space that the rectangle covers. It is important because it will be matched to the area of the square in the next steps of our problem.

Area of a Square

The area of a square can be calculated by squaring the length of one of its sides. A square is a special type of rectangle where all sides are equal in length. This means that if one side is known, the area can be easily determined using the formula.
The formula to find the area is:

• The formula is: \[\text{Area} = s^2\]
• Where \(s\) is the length of a side of the square.

In this problem, we want our square to have the same area as the rectangle, which is 396 square centimeters.

• So, we set the areas equal: \[s^2 = 396\]
• To find the side length, we take the square root of the area: \[s = \sqrt{396}\]

The new challenge arises in simplifying the square root to find the simplest form of the side length.

Square Root Calculation

Finding the square root of a number involves determining a value that, when multiplied by itself, gives the original number. In our problem, we need to simplify \(\sqrt{396}\) to find the side length of the square.
To simplify, we first perform prime factorization of the number:

• The prime factors of 396 are: \[396 = 2 \times 2 \times 3 \times 3 \times 11\]
• This can be written as: \[(2^2) \times (3^2) \times 11\]

From these prime factors, we identify those that can be taken out of the square root by pairing them:

• Taking pairs out, we get: \[2 \times 3 \text{ from }(2^2) \times (3^2)\]
• Which allows us to simplify the expression to: \[6 \sqrt{11}\]

So, the side length of the square in simplest form is \(6 \sqrt{11} \text{ cm}\). This understanding of prime factorization and simplification is crucial for working with square roots effectively.