Question

Imagine that you have 12 square tiles, each measuring 1 inch on a side. a. In how many different ways can you put all 12 tiles together to make a rectangle? Sketch each possible rectangle. b. Which of your rectangles has the greatest perimeter? What is its perimeter? c. Which of your rectangles has the least perimeter? What is its perimeter?

Short answer

The possible rectangles are 1x12, 2x6, and 3x4. The 1x12 rectangle has the greatest perimeter of 26 inches. The 3x4 rectangle has the least perimeter of 14 inches.

Step-by-step solution

Determine Factors of 12

Identify pairs of integers that multiply to 12 to find the possible dimensions of rectangles. The factors of 12 are: 1, 2, 3, 4, 6, and 12.

List Possible Rectangle Dimensions

Using the factors, the possible rectangle dimensions are 1x12, 2x6, and 3x4.

Sketch Each Possible Rectangle

Draw each possible rectangle:1. 1x12 Rectangle2. 2x6 Rectangle3. 3x4 Rectangle

Calculate Perimeters for Each Rectangle

Use the perimeter formula for a rectangle, which is given by P = 2(l + w), where l is the length and w is the width.For each rectangle:1. 1x12: P = 2(1 + 12) = 26 inches2. 2x6: P = 2(2 + 6) = 16 inches3. 3x4: P = 2(3 + 4) = 14 inches

Identify the Rectangle with the Greatest Perimeter

Compare the perimeters calculated in the previous step. The rectangle with the greatest perimeter is 1x12 with a perimeter of 26 inches.

Identify the Rectangle with the Least Perimeter

Compare all perimeters calculated. The rectangle with the least perimeter is 3x4 with a perimeter of 14 inches.

Key concepts

factors of 12

When you want to determine the different ways to form a rectangle using a fixed number of tiles, like 12 in this example, it's helpful to know the factors of that number. A factor is a number that divides evenly into another number. For the number 12, its factors are 1, 2, 3, 4, 6, and 12.
To see why these are factors, consider:
• 1 x 12 = 12
• 2 x 6 = 12
• 3 x 4 = 12
Understanding these factors helps us find out the possible dimensions of the rectangles. This step is fundamental because it directly influences how you can arrange the tiles.
So, whenever you're tasked with forming rectangles from a fixed total number of tiles, first list all pairs of factors.

rectangle dimensions

Knowing the factors of 12 allows us to determine the different dimensions of rectangles we can make. For 12 tiles, the possible rectangle dimensions are:
• 1 inch by 12 inches
• 2 inches by 6 inches
• 3 inches by 4 inches
These pairs of dimensions (length x width) represent the different ways you can arrange the tiles to form a rectangle. Once you have the dimensions, sketching these rectangles will help you visualize how they look.
Visualization is particularly helpful for understanding the layout and can aid in designing and planning arrangements for other practical applications, such as tiling a floor.

perimeter calculations

Once you have the dimensions of all possible rectangles, the next step involves calculating their perimeters. The perimeter of a rectangle is the total distance around it and can be found using the formula:
\[ P = 2(l + w) \]
where `l` represents the length and `w` represents the width. Now, let's calculate the perimeter for each dimension:
• For a 1x12 rectangle, the perimeter is:
\[ P = 2(1 + 12) = 2(13) = 26 \text{ inches} \]
• For a 2x6 rectangle, the perimeter is:
\[ P = 2(2 + 6) = 2(8) = 16 \text{ inches} \]
• For a 3x4 rectangle, the perimeter is:
\[ P = 2(3 + 4) = 2(7) = 14 \text{ inches} \]
By comparing these perimeters, you can see which rectangle has the greatest perimeter (1x12 with 26 inches) and which has the least perimeter (3x4 with 14 inches).
This approach to calculating perimeter not only helps solve the exercise but also strengthens your understanding of handling various geometrical calculations in future problems.