Optimizing Rope Usage: Forming the Largest Square with Whole Number Sides

Roland is faced with a math problem: using a 15 cm rope to form the largest possible square with whole number sides. To solve this problem, we'll explore the process of determining the side length of the square and the amount of rope that Roland must cut. This step-by-step approach will provide clear insights into the solution and the underlying mathematical principles.

Understanding the Problem

The key to solving this problem is to recognize that the perimeter of a square is equal to four times its side length. The formula for the perimeter of a square is given by:

P 4s

Where:

P represents the perimeter of the square.

s represents the length of each side of the square.

Setting Up the Equation

Roland's goal is to maximize the side length s while ensuring that the perimeter does not exceed the total length of the rope, which is 15 cm. Therefore, we set up the following inequality:

4s ≤ 15

Solving for the Largest Whole Number Side Length

To find the maximum side length s, we divide both sides of the inequality by 4:

s ≤ frac{15}{4} 3.75

Since s must be a whole number, the largest possible value for s is 3 cm.

Calculating the Perimeter of the Square

With the determined side length of 3 cm, we can now calculate the perimeter of the square:

P 4s 4 × 3 12 , text{cm}

Calculating the Amount of Rope to Cut

To find out how much rope Roland must cut, we subtract the perimeter of the square from the original length of the rope:

Length to cut 15 , text{cm} - 12 , text{cm} 3 , text{cm}

Thus, Roland must cut 3 cm of the rope to form the square with the largest possible whole number side length.

Your Turn: Finding the Solution

Let's break down the steps for another example where the total length of the rope is 15 cm and the goal is to form the largest possible square with whole number sides:

Identify the total length of the rope: L 15 cm.

Assume the sides of the square will be x cm each.

Calculate the total required length of the rope to make the square:

P 4x

Divide the total length of the rope by 4 to find the maximum quotient:

L ÷ 4 3.75

Recognize that the largest whole number less than or equal to 3.75 is 3. Therefore, s 3 cm.

Cut the remainder when the rope length is divided by 4:

Length to cut L - (4 × s) 15 - (4 × 3) 15 - 12 3 , text{cm}

The length required to make the square is 12 cm, giving a side length of 3 cm for a total area of 9 cm2.

Ensure the final square has four sides of 3 cm each for a total perimeter of 12 cm.

By following these steps, you can solve similar rope cutting problems and determine the largest possible square with whole number sides.