Rectangle Dimensions: Perimeter 60cm, Base = 2x Height

by Andrew McMorgan 55 views

Hey Plastik Magazine readers! Let's dive into a fun geometry problem today. We're going to figure out the dimensions of a rectangle given its perimeter and a relationship between its base and height. It's like a puzzle, and we're going to solve it together. So grab your thinking caps, and let's get started!

Understanding the Problem

The key to solving any math problem is understanding what it's asking. In this case, we have a rectangle. Remember, a rectangle is a four-sided shape where opposite sides are equal and all angles are right angles (90 degrees). We're given two crucial pieces of information:

  1. The perimeter of the rectangle is 60 centimeters. The perimeter is the total distance around the outside of the rectangle. If you were to walk all the way around the rectangle, you would have walked 60 centimeters.
  2. The base (or length) of the rectangle is twice the height (or width). This means if the height is 'h', then the base is '2h'. This relationship is super important because it allows us to express both the base and the height in terms of a single variable.

Our goal is to find the actual values of the height ('h') and the base ('b') that satisfy these conditions. We're given some answer choices, but we want to understand how to arrive at the correct answer, not just pick one randomly. Let's break down the steps.

Setting Up the Equations

Okay, guys, let's get mathematical! To solve this, we need to translate the word problem into equations. This is where the magic happens. We'll use two key formulas:

  1. Perimeter of a rectangle: The perimeter (P) of a rectangle is calculated as P = 2b + 2h, where 'b' is the base and 'h' is the height. This makes sense because you have two sides of length 'b' and two sides of length 'h'.
  2. Relationship between base and height: We know that the base is twice the height, which we can write as b = 2h.

Now, we can substitute the information we have into these equations. We know P = 60 cm, so our perimeter equation becomes:

60 = 2b + 2h

And we already have our relationship equation:

b = 2h

We have two equations and two unknowns (b and h), which means we can solve for them! This is a classic algebra setup. The next step is to use substitution to solve for one of the variables.

Solving for Height and Base

The best way to tackle this system of equations is using substitution. Since we already have b = 2h, we can substitute '2h' for 'b' in the perimeter equation. This will leave us with an equation with only one variable ('h'), which we can easily solve.

So, let's substitute b = 2h into 60 = 2b + 2h:

60 = 2(2h) + 2h

Now, simplify the equation:

60 = 4h + 2h

Combine like terms:

60 = 6h

To isolate 'h', divide both sides of the equation by 6:

h = 60 / 6

h = 10 cm

Awesome! We've found the height of the rectangle. Now that we know the height, we can easily find the base using the relationship b = 2h:

b = 2 * 10 cm

b = 20 cm

So, the height of the rectangle is 10 cm, and the base is 20 cm. We've cracked the code!

Checking the Answer

Before we celebrate too much, let's make sure our answer makes sense. The best way to do this is to plug our values for 'h' and 'b' back into the original equations and see if they hold true.

First, let's check the perimeter:

P = 2b + 2h

P = 2(20 cm) + 2(10 cm)

P = 40 cm + 20 cm

P = 60 cm

Great! The perimeter is indeed 60 cm, as given in the problem. Now, let's check the relationship between the base and the height:

b = 2h

20 cm = 2 * 10 cm

20 cm = 20 cm

This also checks out! Our answer satisfies both conditions of the problem. We can be confident that our solution is correct.

Identifying the Correct Option

Now that we've calculated the dimensions, let's look back at the answer choices provided and identify the correct one.

We found that the height (h) is 10 cm and the base (b) is 20 cm. Looking at the options:

a. h = 9 cm, b = 18 cm b. h = 19 cm, b = 38 cm c. h = 10 cm, b = 20 cm d. h = 10 cm, b = 15 cm

Option (c) matches our calculated dimensions perfectly! So, the correct answer is:

c. h = 10 cm, b = 20 cm

We did it, guys! We successfully solved the problem by understanding the concepts, setting up equations, and carefully working through the steps. Geometry problems like these might seem intimidating at first, but with a little bit of algebra and a clear understanding of the formulas, they become much more manageable.

Key Takeaways

Before we wrap up, let's quickly recap the key takeaways from this problem. Remember these points, and you'll be well-equipped to tackle similar problems in the future:

  • Understand the Problem: The first step is always to carefully read and understand what the problem is asking. Identify the given information and what you need to find.
  • Translate to Equations: Convert the word problem into mathematical equations. This is where you use formulas and relationships to express the problem in a symbolic form.
  • Solve the Equations: Use algebraic techniques, such as substitution or elimination, to solve for the unknown variables.
  • Check Your Answer: Always plug your solution back into the original equations to verify that it satisfies all the conditions of the problem. This is a crucial step to avoid errors.
  • Relate Back to the Context: Make sure your answer makes sense in the context of the problem. For example, if you're finding dimensions, the values should be positive and reasonable.

Geometry problems often involve real-world applications, so understanding how to solve them is not just about getting the right answer; it's about developing problem-solving skills that you can use in many situations. Keep practicing, and you'll become a geometry whiz in no time!

Practice Makes Perfect

Now that we've worked through this problem together, the best way to solidify your understanding is to practice more problems! Try to find similar problems in your textbook or online and work through them step by step. Don't be afraid to make mistakes – that's how you learn. And remember, if you get stuck, go back to the key steps we discussed: understand the problem, set up the equations, solve them, and check your answer.

Geometry is all about shapes and relationships, so the more you practice, the better you'll become at visualizing the problems and finding the solutions. Keep up the great work, Plastik Magazine readers, and I'll see you in the next math adventure!

So, that's it for today's math lesson, folks! I hope you found this helpful and that you're feeling more confident about solving rectangle problems. Remember, math can be fun and rewarding, especially when you break it down step by step. Until next time, keep those brains buzzing!