Calculating Rectangle Sides With A Perimeter Of 40cm
Hey everyone! Let's dive into a fun math problem where we need to figure out the sides of a rectangle. We know the perimeter and a little something about how the base and height relate. Sounds like a puzzle, right? Let’s break it down step by step!
Understanding the Problem
So, the problem states: Calculate the sides of a rectangle where the perimeter is 40cm, and the base exceeds the height by 4cm. To kick things off, we need to remember a few key things about rectangles and their perimeters. A rectangle has two pairs of equal sides – the base and the height. The perimeter is the total distance around the rectangle, which we find by adding up all the sides. In mathematical terms, if we let 'b' represent the base and 'h' represent the height, the perimeter (P) can be calculated using the formula: P = 2b + 2h. This formula is our starting point, but we also have another crucial piece of information: the base exceeds the height by 4cm. This means that the base (b) is 4cm longer than the height (h), which we can write as b = h + 4. Now we have two equations and two unknowns, which means we’re in business! We can use these equations to solve for the base and height. Think of it like fitting pieces of a puzzle together. We have the total perimeter (40cm) and the relationship between the base and height (b = h + 4). Our mission is to use these clues to uncover the exact lengths of each side. It's all about translating the words of the problem into mathematical expressions and then using those expressions to find our answers. Math problems like this are not just about numbers; they're about understanding relationships and using logic to solve mysteries. So, are you ready to put on your detective hats and find those missing sides? Let’s move on to the next section where we’ll start plugging in the numbers and cracking the code!
Setting Up the Equations
Alright, guys, let's get down to the nitty-gritty and set up the equations we'll use to solve this rectangle riddle. As we discussed, we have two vital pieces of information. First, we know the perimeter of the rectangle is 40cm. Remember, the formula for the perimeter (P) of a rectangle is P = 2b + 2h, where 'b' is the base and 'h' is the height. So, we can write our first equation as: 2b + 2h = 40. This equation tells us that if we double the base and double the height, the total will be 40cm. Pretty straightforward, right? But we have two unknowns (b and h), and we need another equation to solve for them. This is where our second piece of information comes in handy. We know that the base exceeds the height by 4cm. In mathematical language, this translates to: b = h + 4. This equation is super useful because it tells us exactly how the base and height are related. It says the base is simply the height plus 4cm. Now we have a system of two equations:
- 2b + 2h = 40
- b = h + 4
This is where the magic happens! We can use these equations together to find the values of 'b' and 'h'. There are a couple of ways to approach this. One common method is substitution, where we substitute the value of 'b' from the second equation into the first equation. This will leave us with an equation that only has one unknown (h), which we can easily solve. Another method is elimination, but substitution is often the simpler route for this type of problem. The key here is to see how we've transformed the word problem into a set of mathematical equations. This is a crucial skill in algebra and problem-solving in general. Once we have these equations, the rest is just algebraic manipulation. So, let's roll up our sleeves and get ready to solve these equations in the next section. We're one step closer to uncovering the mystery of the rectangle's sides!
Solving the Equations
Okay, team, time to put our math skills to the test and solve these equations! We've got our system of equations ready to go:
- 2b + 2h = 40
- b = h + 4
As we discussed, the substitution method is a great way to tackle this. We'll take the second equation (b = h + 4) and substitute 'b' in the first equation with 'h + 4'. This means wherever we see 'b' in the first equation, we'll replace it with 'h + 4'. Let's do it! Our first equation, 2b + 2h = 40, becomes: 2(h + 4) + 2h = 40. See what we did there? We swapped 'b' for 'h + 4'. Now, we need to simplify this equation. First, we distribute the 2 across the parentheses: 2h + 8 + 2h = 40. Next, we combine like terms (the 'h' terms): 4h + 8 = 40. Now we're getting somewhere! We want to isolate 'h' on one side of the equation. To do that, we subtract 8 from both sides: 4h = 32. Finally, we divide both sides by 4 to solve for 'h': h = 8. Woohoo! We found the height! The height of the rectangle is 8cm. But we're not done yet. We still need to find the base. Luckily, we have the second equation (b = h + 4) to help us. We know 'h' is 8, so we can plug that in: b = 8 + 4. This gives us: b = 12. So, the base of the rectangle is 12cm. We've done it! We've successfully solved for both the base and the height. But before we celebrate too much, let's double-check our work to make sure everything makes sense. In the next section, we'll verify our answers and make sure they fit the original problem.
Verifying the Solution
Alright, math detectives, it's time to put on our verification hats and make sure our solution holds up! We found that the height (h) of the rectangle is 8cm and the base (b) is 12cm. Now, we need to check if these values satisfy the conditions given in the original problem. First, let's check the perimeter. We know the perimeter should be 40cm. Using our values, the perimeter is 2b + 2h = 2(12) + 2(8) = 24 + 16 = 40cm. Awesome! Our values satisfy the perimeter condition. Next, we need to check if the base exceeds the height by 4cm. Our base is 12cm and our height is 8cm. Is 12cm 4cm more than 8cm? Yes, it is! So, our values also satisfy the condition that the base exceeds the height by 4cm. We've successfully verified that our solution is correct! The height of the rectangle is 8cm, and the base is 12cm. This is a crucial step in problem-solving. It's not enough to just find an answer; you need to make sure it makes sense in the context of the problem. Verifying your solution helps you catch any errors and builds confidence in your answer. Think of it like proofreading your work before you submit it. It's a small step that can make a big difference. So, we've solved the mystery of the rectangle's sides! We used our knowledge of perimeters, set up equations, solved for the unknowns, and verified our solution. That's some serious math detective work! Now, let's wrap things up with a final summary of our findings in the next section.
Final Answer
Okay, everyone, let's bring it all together and present our final answer! After carefully working through the problem, setting up our equations, solving for the unknowns, and verifying our solution, we've successfully determined the dimensions of the rectangle. The height of the rectangle is 8cm, and the base of the rectangle is 12cm. We found these values by using the information given in the problem – the perimeter is 40cm, and the base exceeds the height by 4cm. We translated these pieces of information into mathematical equations, used the substitution method to solve for the unknowns, and then verified our solution to ensure it was accurate. This problem is a great example of how we can use algebra to solve real-world problems. It demonstrates the power of translating word problems into mathematical expressions and then using those expressions to find solutions. It also highlights the importance of verifying your work to ensure accuracy. So, there you have it! The sides of the rectangle are 8cm and 12cm. We solved it! Give yourselves a pat on the back for your awesome math skills!
