Rectangle Area And Perimeter: A Math Challenge

by TextBrain Team 47 views

Hey guys! Let's dive into a fun math problem. We've got a rectangle, and we know its area is 42 cm². One of the sides is a prime number, and our mission? To figure out which of the provided options cannot be the perimeter of this rectangle. Sound like a plan? Let's get started!

Understanding the Problem

Okay, so we're dealing with a rectangle. Remember what we know about rectangles? They have four sides, with opposite sides being equal in length. The area of a rectangle is calculated by multiplying its length and width (Area = Length x Width). The perimeter, on the other hand, is the total distance around the outside, found by adding up the lengths of all four sides (Perimeter = 2 x (Length + Width)).

We're also told that one of the sides has a length that is a prime number. Prime numbers are whole numbers greater than 1 that are only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, and so on. This prime number constraint is super important, so keep that in mind! The area of our rectangle is 42 cm². This information gives us a direct relationship between the length and width since we know that Length * Width = 42.

To tackle this problem, we'll need to consider different pairs of numbers that multiply to give us 42, and then check which of these pairs includes a prime number. Once we've identified these valid length and width pairs, we'll calculate the perimeter for each and compare it to the options provided to see which one doesn't fit. It's all about using the properties of rectangles, prime numbers, area, and perimeter to crack the code. This problem isn't just about math; it's about logical thinking and problem-solving!

Finding the Possible Dimensions and Perimeters

Alright, let's get down to business and figure out the possible dimensions of our rectangle, keeping in mind that one side must be a prime number. Since the area is 42 cm², we need to find pairs of factors (numbers that multiply together to get 42). Here's how we can break it down:

  • 1 and 42: 1 isn't a prime number, so this pair doesn't work for us.
  • 2 and 21: 2 is a prime number! This pair is a potential candidate. If one side is 2 cm and the other is 21 cm, then the perimeter would be 2 * (2 + 21) = 2 * 23 = 46 cm.
  • 3 and 14: 3 is a prime number. This is another possible combination. If one side is 3 cm and the other is 14 cm, the perimeter would be 2 * (3 + 14) = 2 * 17 = 34 cm.
  • 6 and 7: 7 is a prime number! This pair is also promising. If one side is 6 cm and the other is 7 cm, then the perimeter would be 2 * (6 + 7) = 2 * 13 = 26 cm.

So, we've identified three possible sets of dimensions where one side is a prime number, and we've calculated the perimeters for each: 46 cm, 34 cm, and 26 cm. Now we can look at the answer choices and see which one cannot be the perimeter of the rectangle.

Checking the Answer Choices

We've crunched the numbers and determined the perimeters we can achieve given the constraints of our rectangle problem, including one prime side. Let's see how these results stack up against the answer choices:

  • A) 26: We found that a perimeter of 26 cm is possible (with sides of 6 cm and 7 cm). So, this is a viable option.
  • B) 34: We also found a perimeter of 34 cm (with sides of 3 cm and 14 cm). This is a valid perimeter.
  • C) 46: Our calculations showed that a perimeter of 46 cm is possible (using sides of 2 cm and 21 cm). This checks out.
  • D) 86: Hmm, we didn't find a perimeter of 86 cm with any of the prime number side combinations. Let's check. If the perimeter is 86, then half the perimeter (which equals the sum of the length and width) would be 43 cm. For two numbers to add up to 43, and one of them to be a factor of 42, the only factors are 1, 2, 3, 6, 7, 14, 21, and 42. There's no way for us to get 43 as a sum of two factors of 42 where one factor is a prime number. Therefore, 86 is not possible. So, the answer that cannot be the perimeter of this rectangle is 86.

Conclusion

Therefore, the answer is D) 86. The perimeter of the rectangle cannot be 86 cm, given that one of its sides is a prime number and its area is 42 cm². We systematically considered the factors of 42, identified the pairs with a prime number, calculated their perimeters, and compared them to the answer choices to eliminate incorrect ones. Way to go, everyone! This shows how you can apply your understanding of geometric shapes, factors, prime numbers, and perimeters to solve mathematical puzzles. Keep practicing, and you'll get better and better at this stuff!

Additional Tips for Similar Problems:

  • Always List Factors: When dealing with area problems, start by listing all the factor pairs of the given area. This ensures that you don't miss any possible dimensions.
  • Identify Constraints: Pay close attention to any special conditions, like a prime number requirement, as these are crucial for narrowing down the possible solutions.
  • Draw a Diagram: Visualizing the problem with a simple sketch of the rectangle can help you stay organized and avoid mistakes.
  • Check Your Work: Once you think you've found the answer, double-check your calculations and make sure they align with all the given conditions. This helps catch any errors early on.
  • Practice Makes Perfect: The more you practice these types of problems, the quicker and more confident you'll become at solving them. Don't be afraid to try different examples and challenge yourself!