Rectangle's Perimeter: Area 84cm², Unveiling Min & Max Values

Hey math enthusiasts! Let's dive into a fun geometry problem. We're talking about a rectangle, that classic shape we all know and love, and we're going to explore its perimeter based on its area. Specifically, we're focusing on a rectangle where the sides are whole numbers (natural numbers, as mathematicians like to say), and its area is exactly 84 square centimeters. Our mission? To find the smallest and largest possible perimeters this rectangle can have. Plus, we'll investigate what happens when the perimeter is specifically 40 cm. Ready to get started, guys?

Understanding the Basics: Area, Perimeter, and Rectangles

Before we jump into the calculations, let's refresh our memories on the fundamental concepts. A rectangle is a four-sided shape with four right angles (90 degrees each). Opposite sides of a rectangle are equal in length. The area of a rectangle is the space it occupies, calculated by multiplying its length and width: Area = Length × Width. The perimeter, on the other hand, is the total distance around the rectangle, found by adding up the lengths of all its sides: Perimeter = 2 × (Length + Width). Now, in our specific scenario, we're given that the area is 84 cm². This means that when we multiply the length and width of our rectangle, the result must always be 84. And, since the sides are natural numbers, we're looking for whole-number factors of 84. These factors will represent the different possible lengths and widths of our rectangle. Let's find these factors, which will then help us to figure out the perimeter, ultimately assisting us in finding the minimum and maximum perimeter.

To find the factors of 84, let's systematically go through the numbers. We can start with 1 and go up, checking if each number divides 84 without a remainder. Here's a table to organize our findings:

We don't need to go further, because after 7, we will start repeating the factors (i.e. 12 x 7 = 84, 14 x 6 = 84, and so on). The smallest perimeter we found is 38 cm, and the largest is 170 cm. Note that as the difference between the length and width increases, the perimeter increases too, and vice versa. It's like, a very long and skinny rectangle will have a huge perimeter, while a more square-like rectangle will have a smaller perimeter. This brings us to another interesting part. If the perimeter is 40 cm, we know that the length and width must be 6 cm and 14 cm, as seen in the table above.

Finding the Minimum and Maximum Perimeter

Alright, now that we've refreshed our knowledge and found some possibilities, let's get down to brass tacks: determining the smallest and largest possible perimeters. To find the minimum perimeter, we need to find the combination of length and width that are closest in value. Think of it like trying to make the rectangle as close to a square as possible while still having an area of 84 cm². Looking at our table of factors, the pair that gives us the smallest perimeter is 7 cm (length) and 12 cm (width). Thus, the minimum perimeter is 2 × (7 + 12) = 38 cm.

On the other hand, to find the maximum perimeter, we want the length and width to be as different as possible. This means one side should be very long and the other very short, resulting in a long, skinny rectangle. Again, from our factor pairs, we can see that the pair that gives us the largest perimeter is 1 cm (length) and 84 cm (width). Therefore, the maximum perimeter is 2 × (1 + 84) = 170 cm. So, the smallest value for the perimeter is 38 cm, and the largest value is 170 cm. Got it, guys?

Let's summarize, in a more formal way:

The smallest perimeter occurs when the rectangle is closest to a square. The dimensions are 7 cm and 12 cm, and the perimeter is 38 cm. The largest perimeter occurs when the rectangle is most elongated. The dimensions are 1 cm and 84 cm, and the perimeter is 170 cm.

The Special Case: Perimeter of 40 cm

Now, what if the perimeter is specifically 40 cm? This is a bit like a reverse engineering problem. We know the perimeter and need to figure out the dimensions. We know that Perimeter = 2 × (Length + Width), so if the perimeter is 40 cm, then 40 = 2 × (Length + Width). Dividing both sides by 2, we get 20 = Length + Width. This means that the length and width must add up to 20. But, they also need to multiply to give us an area of 84. Looking back at our factor pairs, we can identify which pair adds up to give a perimeter of 40 cm. The only pair that fits is 6 cm (length) and 14 cm (width), because 6 x 14 = 84, and 2 x (6 + 14) = 40. Now we know, if this rectangle has a perimeter of 40 cm, then its sides must be 6 cm and 14 cm.

Conclusion: Perimeter Power-Up!

So, there you have it, folks! We've successfully navigated the world of rectangles, areas, and perimeters. We've discovered the minimum and maximum possible perimeters for a rectangle with an area of 84 cm², and explored a special case where the perimeter is specifically 40 cm. The key takeaway is the relationship between the dimensions of a rectangle and its perimeter: the closer the dimensions are, the smaller the perimeter; the more they differ, the larger the perimeter. It is important to know all of the rules.

This kind of problem helps us think about different shapes and the mathematical relationships between their properties. Keep up the good work and keep exploring the amazing world of mathematics! Until next time, keep calculating and stay curious!