Minimum Perimeter: Rectangle With 12 Cm² Area

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Hey guys! Today, we're diving into a fun geometry problem that combines a bit of number theory with our knowledge of rectangles. We're going to figure out how to find the smallest possible perimeter of a rectangle when we know its area and that its sides have a special relationship. So, let's jump right in and explore this cool problem together!

Understanding the Problem

Okay, so the problem states that we have a rectangle whose area is exactly 12 square centimeters. That's our key piece of information! But there’s another crucial detail: the side lengths of this rectangle, measured in centimeters, are coprime. Now, what does coprime mean? Well, it's a fancy way of saying that the two numbers representing the side lengths have no common factors other than 1. In simpler terms, they can't both be divided evenly by the same number (other than 1, of course). For example, 3 and 4 are coprime, but 4 and 6 are not (because they can both be divided by 2). Our mission, should we choose to accept it (and we do!), is to find the smallest possible perimeter for such a rectangle. Remember, the perimeter is the total distance around the outside of the rectangle – we get it by adding up the lengths of all four sides.

To really nail this, it's essential to visualize what we're dealing with. Imagine different rectangles, all squished and stretched in various ways, but each one perfectly enclosing an area of 12 cm². Some might be long and skinny, others closer to squares. The coprime condition limits our options, as only certain combinations of side lengths will fit the bill. We need to systematically explore these possibilities to pinpoint the one that gives us the absolute minimum perimeter. We aren't just looking for any old rectangle with an area of 12 cm²; we're on a quest for the one with the smallest 'fence' around it, given the coprime constraint. This blend of geometry and number theory makes the problem a fun brain-teaser, perfect for sharpening our problem-solving skills.

Thinking about the formula for the area of a rectangle (Area = length × width) is super helpful here. We need to find pairs of numbers that multiply to 12, and then check which of those pairs are coprime. Once we have those coprime pairs, we can calculate the perimeter for each (Perimeter = 2 × (length + width)) and compare them. This step-by-step approach will guide us to the solution. So, let's get started by listing out the factor pairs of 12!

Finding Coprime Side Lengths

Alright, let's get down to business and figure out those coprime side lengths. The area of a rectangle, as we all know, is calculated by multiplying its length and width. So, we need to find pairs of whole numbers that multiply together to give us 12. These pairs are called factors or divisors of 12. Let’s list them out systematically to make sure we don't miss any:

  • 1 and 12 (1 x 12 = 12)
  • 2 and 6 (2 x 6 = 12)
  • 3 and 4 (3 x 4 = 12)

Now, we have three potential pairs of side lengths. But remember, there's that crucial condition: the side lengths have to be coprime. This means they shouldn't share any common factors other than 1. So, let’s examine each pair and see if they meet this requirement.

  • 1 and 12: The factors of 1 are just 1. The factors of 12 are 1, 2, 3, 4, 6, and 12. The only common factor is 1. So, 1 and 12 are coprime!
  • 2 and 6: The factors of 2 are 1 and 2. The factors of 6 are 1, 2, 3, and 6. They share a common factor of 2 (in addition to 1). So, 2 and 6 are not coprime.
  • 3 and 4: The factors of 3 are 1 and 3. The factors of 4 are 1, 2, and 4. The only common factor is 1. So, 3 and 4 are coprime!

Great! We've narrowed it down. We have two pairs of side lengths that fit the coprime condition: 1 and 12, and 3 and 4. Now, to find the smallest perimeter, we need to calculate the perimeter for each of these rectangles and compare them. It's like we're designing two different rectangles with the same area but different shapes, and we want to see which one needs the least amount of fencing to enclose it. This is where the perimeter formula comes into play, so let's use that next!

Calculating the Perimeters

Okay, so we've identified the two pairs of coprime side lengths that give us an area of 12 cm²: 1 cm and 12 cm, and 3 cm and 4 cm. Now comes the moment of truth – let's calculate the perimeter for each of these rectangles and see which one has the smallest perimeter. Remember, the formula for the perimeter of a rectangle is: Perimeter = 2 × (length + width).

Let's start with the first pair:

  • Rectangle 1: Length = 1 cm, Width = 12 cm
    • Perimeter = 2 × (1 cm + 12 cm)
    • Perimeter = 2 × (13 cm)
    • Perimeter = 26 cm

So, a rectangle with sides 1 cm and 12 cm has a perimeter of 26 cm. That's one contender for the smallest perimeter. Now, let's calculate the perimeter for the second rectangle:

  • Rectangle 2: Length = 3 cm, Width = 4 cm
    • Perimeter = 2 × (3 cm + 4 cm)
    • Perimeter = 2 × (7 cm)
    • Perimeter = 14 cm

Aha! A rectangle with sides 3 cm and 4 cm has a perimeter of 14 cm. Now we can directly compare the two perimeters we've calculated. We have 26 cm for the first rectangle and 14 cm for the second. It's clear that 14 cm is smaller than 26 cm. This means that the rectangle with sides 3 cm and 4 cm has the smallest perimeter among all rectangles with an area of 12 cm² and coprime side lengths. Woohoo! We're almost there – we've done the heavy lifting of finding the coprime pairs and calculating the perimeters. The final step is to state our answer clearly, making sure we've answered the question precisely.

Determining the Minimum Perimeter

Alright, drumroll please! We've done the detective work, crunched the numbers, and now it's time to reveal the answer. We calculated the perimeters for the two possible rectangles with an area of 12 cm² and coprime side lengths:

  • Rectangle with sides 1 cm and 12 cm: Perimeter = 26 cm
  • Rectangle with sides 3 cm and 4 cm: Perimeter = 14 cm

By comparing these two values, it's crystal clear that the rectangle with sides 3 cm and 4 cm has the smallest perimeter. Therefore, the minimum perimeter for a rectangle with an area of 12 cm² and coprime side lengths is 14 cm. That's our final answer!

So, to wrap it up, when we're dealing with problems that involve minimizing perimeters or maximizing areas, it's crucial to consider all the constraints given in the problem. In this case, the coprime condition was the key that unlocked the solution. It narrowed down our possibilities and allowed us to pinpoint the specific rectangle with the minimum perimeter. This type of problem beautifully illustrates how different areas of math, like geometry and number theory, can come together to solve interesting challenges. It's not just about memorizing formulas; it's about thinking critically and creatively to find the best solution. And you guys absolutely nailed it! Pat yourselves on the back for conquering this geometry puzzle!

Conclusion

So, there you have it! The smallest perimeter for a rectangle with an area of 12 cm² and coprime side lengths is a neat and tidy 14 cm. This problem wasn't just about knowing the formulas for area and perimeter; it was also about understanding what coprime numbers are and how they can influence geometric shapes. By systematically listing the factors of 12, identifying the coprime pairs, and then calculating the perimeters, we were able to arrive at the solution. This kind of problem is a fantastic way to flex your mathematical muscles and see how different concepts can be interconnected. Keep practicing, keep exploring, and most importantly, keep having fun with math! You guys are doing awesome!