Rectangle Side Length Problem: Math Solution

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Hey guys! Today, we're diving into a cool math problem that involves rectangles and their areas. We've got two rectangular pieces of paper, and we're trying to figure out what happens when we stick them together. Let's break it down step by step so it's super clear.

Understanding the Problem

So, here’s the deal. We have two rectangles. The first rectangle has an area of 24 square millimeters, and the second one has an area of 42 square millimeters. All the sides of these rectangles are whole numbers (integers) when measured in millimeters. Now, we're taking the shorter sides of these rectangles and joining them together. The question we need to answer is: what is the length of the longer side of the new rectangle we've created?

This might sound a bit confusing at first, but don't worry! We'll untangle it. The key here is to remember what area means and how the sides of a rectangle relate to it. The area of a rectangle is simply its length multiplied by its width. So, to solve this, we need to think about all the possible lengths and widths that give us areas of 24 and 42.

Why is this important? Well, this type of problem tests our understanding of factors (numbers that divide evenly into another number) and how they relate to geometry. It’s not just about formulas; it’s about logical thinking and problem-solving skills. Plus, it’s a great example of how math pops up in real-life situations, even when we're just dealing with pieces of paper!

Finding Possible Side Lengths

Okay, so the first step is to figure out the possible side lengths for each rectangle. Remember, the area of a rectangle is calculated by multiplying its length and width (Area = Length × Width). Since we know the areas, we need to find pairs of numbers that multiply to give us 24 and 42. These pairs of numbers are called factors.

Rectangle 1: Area = 24 square millimeters

Let's list the pairs of whole numbers that multiply to 24:

  • 1 × 24 = 24
  • 2 × 12 = 24
  • 3 × 8 = 24
  • 4 × 6 = 24

So, the possible dimensions (length and width) for the first rectangle are 1 mm by 24 mm, 2 mm by 12 mm, 3 mm by 8 mm, and 4 mm by 6 mm. We’ve got four options to work with here.

Rectangle 2: Area = 42 square millimeters

Now, let’s do the same for the second rectangle, which has an area of 42 square millimeters. We need to find pairs of numbers that multiply to 42:

  • 1 × 42 = 42
  • 2 × 21 = 42
  • 3 × 14 = 42
  • 6 × 7 = 42

For the second rectangle, the possible dimensions are 1 mm by 42 mm, 2 mm by 21 mm, 3 mm by 14 mm, and 6 mm by 7 mm. Again, we have four possible sets of dimensions.

Why are we doing this? By finding all the possible dimensions, we can figure out which shorter sides can be joined together, and what the resulting longer side will be. It’s like having puzzle pieces – we need to see which ones fit!

Combining the Rectangles

Alright, we've got our possible side lengths for both rectangles. Now comes the fun part: figuring out what happens when we join the shorter sides together. Remember, the problem asks for the length of the longer side of the new rectangle formed. So, we need to carefully consider the dimensions.

First, let’s identify the shorter sides for each possible dimension:

  • Rectangle 1:
    • 1 mm × 24 mm (Shorter side: 1 mm)
    • 2 mm × 12 mm (Shorter side: 2 mm)
    • 3 mm × 8 mm (Shorter side: 3 mm)
    • 4 mm × 6 mm (Shorter side: 4 mm)
  • Rectangle 2:
    • 1 mm × 42 mm (Shorter side: 1 mm)
    • 2 mm × 21 mm (Shorter side: 2 mm)
    • 3 mm × 14 mm (Shorter side: 3 mm)
    • 6 mm × 7 mm (Shorter side: 6 mm)

When we join the shorter sides, we're essentially adding the longer sides together to form the new longer side. So, we need to look at all the possible combinations of shorter sides and see what lengths we get when we add the corresponding longer sides.

Why is this step crucial? This is where we really use our problem-solving skills. We're not just plugging numbers into a formula; we're visualizing how these rectangles fit together and thinking about the different possibilities.

Finding the Maximum Length

Now, let's systematically go through the combinations and calculate the lengths of the new longer sides. This is where things might seem a bit tedious, but it's important to be thorough to make sure we find the correct answer.

We'll pair each shorter side from Rectangle 1 with each shorter side from Rectangle 2 and calculate the new longer side:

  • 1 mm (Rect. 1) + 1 mm (Rect. 2): 24 mm + 42 mm = 66 mm
  • 1 mm (Rect. 1) + 2 mm (Rect. 2): 24 mm + 21 mm = 45 mm
  • 1 mm (Rect. 1) + 3 mm (Rect. 2): 24 mm + 14 mm = 38 mm
  • 1 mm (Rect. 1) + 6 mm (Rect. 2): 24 mm + 7 mm = 31 mm
  • 2 mm (Rect. 1) + 1 mm (Rect. 2): 12 mm + 42 mm = 54 mm
  • 2 mm (Rect. 1) + 2 mm (Rect. 2): 12 mm + 21 mm = 33 mm
  • 2 mm (Rect. 1) + 3 mm (Rect. 2): 12 mm + 14 mm = 26 mm
  • 2 mm (Rect. 1) + 6 mm (Rect. 2): 12 mm + 7 mm = 19 mm
  • 3 mm (Rect. 1) + 1 mm (Rect. 2): 8 mm + 42 mm = 50 mm
  • 3 mm (Rect. 1) + 2 mm (Rect. 2): 8 mm + 21 mm = 29 mm
  • 3 mm (Rect. 1) + 3 mm (Rect. 2): 8 mm + 14 mm = 22 mm
  • 3 mm (Rect. 1) + 6 mm (Rect. 2): 8 mm + 7 mm = 15 mm
  • 4 mm (Rect. 1) + 1 mm (Rect. 2): 6 mm + 42 mm = 48 mm
  • 4 mm (Rect. 1) + 2 mm (Rect. 2): 6 mm + 21 mm = 27 mm
  • 4 mm (Rect. 1) + 3 mm (Rect. 2): 6 mm + 14 mm = 20 mm
  • 4 mm (Rect. 1) + 6 mm (Rect. 2): 6 mm + 7 mm = 13 mm

Now, we look through this list and find the largest value. It's 66 mm!

Why go through all these combinations? It’s tempting to jump to conclusions, but math is about being precise. By checking every possible combination, we make sure we haven’t missed the actual maximum length.

The Answer

So, after all that calculation, we've got our answer! The maximum possible length of the longer side of the new rectangle is 66 millimeters. Woohoo! We did it!

Key takeaway: This problem demonstrates how understanding factors and systematically exploring possibilities can help us solve geometric problems. It's not just about knowing the formulas; it's about thinking critically and logically.

Wrapping Up

This rectangle problem might have seemed a bit tricky at first, but by breaking it down into smaller steps – finding factors, identifying shorter sides, and calculating combinations – we were able to find the solution. Remember, in math, it’s all about taking complex problems and making them manageable. Keep practicing, and you'll become a pro at these types of challenges!