Rectangle Area Problem: Prime Factors And Dimensions

Hey guys! Today, we're diving into a cool math problem that involves rectangles, prime factors, and a little bit of geometry. Let's break it down step by step so we can all understand how to solve it. So, buckle up, and let’s get started!

Understanding the Problem

So, here’s the deal: We have a rectangle. This rectangle has a long side that measures 42 cm. Now, the interesting part is the short side. The length of this short side is equal to the sum of the distinct prime factors of the long side’s length. We need to figure out what the area of one face of this rectangle is, and we need to give our answer in cm². Sounds like a plan? Let's dive in!

To solve this problem effectively, we need to understand a few key concepts. First, let’s talk about rectangles. A rectangle is a four-sided shape with opposite sides that are equal in length and four right angles (90-degree angles). The area of a rectangle is calculated by multiplying its length by its width. In our case, we know the length (42 cm), but we need to find the width (the short side). This is where prime factors come into play. Understanding prime factorization is crucial. Prime factorization is breaking down a number into its prime factors. A prime number is a number that has only two divisors: 1 and itself. For example, 2, 3, 5, 7, and 11 are prime numbers. To find the prime factors of 42, we need to break it down into a product of prime numbers. This involves systematically dividing the number by prime numbers until we are left with only prime factors. Finally, we need to understand what distinct prime factors are. The word “distinct” simply means unique or different. So, if a prime factor appears more than once in the prime factorization, we only count it once when we are summing them up to find the length of the short side. By understanding these concepts, we're well-equipped to tackle the problem. Let's move on to the next step and apply these ideas to find our solution. Are you ready? Let’s go!

Finding the Prime Factors of 42

Alright, let’s get our hands dirty and find those prime factors! We need to break down 42 into its prime number building blocks. Remember, prime numbers are numbers that can only be divided by 1 and themselves. So, we're talking about numbers like 2, 3, 5, 7, 11, and so on. We're on a mission to express 42 as a product of these primes.

So, how do we do this? We start by dividing 42 by the smallest prime number, which is 2. Can 42 be divided evenly by 2? You bet! 42 ÷ 2 = 21. Great, so we've got our first prime factor: 2. Now we're left with 21. Let’s keep going. Next up, we try the next prime number, which is 3. Can 21 be divided evenly by 3? Absolutely! 21 ÷ 3 = 7. Fantastic, we've found another prime factor: 3. Now we're down to 7. Is 7 a prime number? You know it! 7 can only be divided by 1 and 7. So, we've hit the jackpot. Our final prime factor is 7.

So, what does this all mean? It means we can express 42 as a product of its prime factors: 42 = 2 × 3 × 7. These are the prime building blocks of 42. We've successfully broken down 42 into its prime factors. Now, remember the problem statement? It mentioned something about the distinct prime factors. In our case, the prime factors 2, 3, and 7 are all different, so they are all distinct. There are no repeats here. This makes our job a bit easier. Understanding how to find these prime factors is super important for solving the rest of the problem. We now know the prime factors of 42. The next step is to use these factors to find the length of the short side of the rectangle. Let's jump into that now and keep the momentum going!

Calculating the Short Side Length

Okay, guys, we've cracked the code on the prime factors of 42. We know that 42 can be broken down into 2 × 3 × 7. Now, let's circle back to the original problem. Remember, the length of the short side of our rectangle is equal to the sum of the distinct prime factors of 42. This is a crucial piece of information that will help us find the short side's length. So, what do we need to do? We need to add up those distinct prime factors we just found.

We identified the distinct prime factors as 2, 3, and 7. Now, let's add them together: 2 + 3 + 7. What do we get? If you add 2 and 3, you get 5. Then, if you add 7 to 5, you get 12. So, the sum of the distinct prime factors of 42 is 12. This is a significant step! What does this 12 represent? According to the problem, this is the length of the short side of our rectangle. We've successfully calculated the length of the short side. It's 12 cm. We now know both the long side (42 cm) and the short side (12 cm) of our rectangle. This is fantastic progress! We're one step closer to solving the problem. Now that we have both dimensions, we can easily calculate the area of the rectangle. Understanding how to use prime factors to find the dimensions is key here. We've done the hard work of finding the prime factors and summing them up. Next, we're going to put these pieces together to find the final answer. Are you ready to calculate the area? Let’s jump to it!

Determining the Area of the Rectangle

Alright, let's bring it all home and calculate the area of our rectangle. We’ve done the groundwork, found the prime factors, and figured out the length of the short side. Now, it’s time to put those numbers to work. Remember, the area of a rectangle is found by multiplying its length by its width. In our case, the length (the long side) is 42 cm, and the width (the short side) is 12 cm. So, to find the area, we need to multiply these two numbers together.

Let’s do the math: 42 cm × 12 cm. Grab your calculators or do it the old-fashioned way – whatever works for you! If you multiply 42 by 12, you get 504. So, what does this 504 represent? This is the area of one face of the rectangle. And we need to include the units in our answer. Since we multiplied centimeters by centimeters, the area is in square centimeters (cm²). Therefore, the area of our rectangle is 504 cm². Woohoo! We did it!

We've successfully calculated the area by using the given information and our knowledge of prime factors. This problem is a great example of how different math concepts can come together to solve a real-world problem. We started with prime factorization, used it to find the dimensions of a rectangle, and then calculated the area. It's like a mathematical puzzle where each step builds on the previous one. Understanding how to calculate the area of a rectangle is a fundamental skill in geometry. We’ve not only solved this specific problem but also reinforced this important concept. Now, let’s take a moment to summarize our steps and celebrate our achievement. Are you ready for a quick recap? Let's go!

Final Answer

Alright guys, let’s recap what we've done to solve this problem. We started with a rectangle that had a long side of 42 cm. The key to solving this was understanding that the short side’s length was the sum of the distinct prime factors of 42. So, our first step was to find those prime factors. We broke down 42 into 2 × 3 × 7. These are the prime building blocks of 42, and they’re all distinct, which made things a bit easier for us.

Next, we needed to find the length of the short side. The problem told us it was the sum of these distinct prime factors. So, we added 2 + 3 + 7, which gave us 12 cm. Now we knew both the long side (42 cm) and the short side (12 cm). With the dimensions in hand, we moved on to the final step: calculating the area of the rectangle. We remembered that the area of a rectangle is length times width, so we multiplied 42 cm by 12 cm. This gave us 504 cm². And that’s our final answer! The area of one face of the rectangle is 504 square centimeters. We’ve successfully navigated through the problem, using prime factorization and basic geometry to reach our solution. Great job, everyone!

So, there you have it! We've successfully solved this rectangle problem by breaking it down step by step. Remember, math problems can seem tricky at first, but with a little bit of understanding and a systematic approach, you can conquer them. Keep practicing, and you'll become a math whiz in no time. Until next time, keep those brains buzzing!