Unlocking Cylinder Height: A Math Problem Solved!

by Tim Redaksi 50 views
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Hey guys! Ever stumbled upon a math problem and thought, "Whoa, where do I even begin?" Well, let's break down a classic geometry puzzle: figuring out the height of a cylinder. We've got the surface area and radius, and our mission? To find that elusive height. Ready to dive in? Let's get started!

Understanding the Problem: The Cylinder's Secrets

First things first, let's unpack what we're dealing with. We're given the surface area of a cylinder, which is 120π cm², and the radius of its circular base, which is 5 cm. Our goal? To calculate the cylinder's height. This isn't just about plugging numbers into a formula; it's about understanding how a cylinder's different parts—the top, bottom, and curved side—contribute to its overall surface area. Think of it like this: If you could unwrap the cylinder, you'd have two circles (the top and bottom) and a rectangle (the curved side). The surface area is the total area of all these shapes combined. The surface area of a cylinder is like the total amount of wrapping paper you'd need to cover it completely. The radius is the distance from the center of the circular top or bottom to its edge. And the height? That's the vertical distance between the two circular bases. Got it? Awesome! Let's get down to business.

Now, before we jump into the formula, let's quickly review the components that make up the surface area of a cylinder. You have two circular bases, and a curved rectangular surface that connects them. The surface area formula takes into account the area of these three elements: two circles (top and bottom) and a rectangle (the curved side). Knowing this, we can approach our height calculation with more confidence. The formula acts like a roadmap that will help you go from your starting point (surface area and radius) to your destination (the height).

To find the height, we're essentially working backward from the total surface area. We'll use the surface area formula, plug in what we know, and then solve for the height. This approach not only provides the correct answer but also helps us visualize the cylinder in terms of its parts. By tackling this problem, we're not just finding a numerical value; we're also deepening our understanding of geometric shapes and how their different aspects relate to each other. This is crucial for future math endeavors. So, let's prepare to become math ninjas and unlock the cylinder's secrets!

The Formula: Your Secret Weapon

Alright, time to arm ourselves with the key formula: Surface Area (SA) = 2πr² + 2πrh. Don't worry, it's not as scary as it looks! Let's break it down:

  • SA: Surface Area (the total area we're working with).
  • r: Radius (the distance from the center of the circular base to its edge).
  • h: Height (what we're trying to find! The vertical distance).
  • Ï€ (Pi): A constant, approximately equal to 3.14159 (but we'll often just use Ï€ in our calculations to keep things clean).

This formula is like a recipe. You put in the ingredients (radius and surface area), and out comes the result (height). The formula is the secret code we'll use to crack the problem. It neatly combines all the areas – the top circle, the bottom circle, and the curved side – into one handy equation. It's the ultimate tool for solving this particular challenge.

Here’s how we'll use it: We know the surface area (120π cm²) and the radius (5 cm). We'll plug these values into the formula and rearrange it to solve for 'h' (height). Then, it's just a matter of doing the math and voilà! The height will magically appear. Remember, understanding the formula is half the battle. This equation is the core of our solution; it enables us to connect the known elements (SA and r) with the unknown (h). Think of the formula as your compass in the geometric landscape; it always points you in the right direction. The more you familiarize yourself with the equation, the easier and more intuitive the process becomes. Using formulas effectively is a key skill in mathematics and in real-life problem-solving.

Plugging in the Numbers and Solving for Height

Okay, time to get our hands dirty (figuratively, of course!). We'll substitute the known values into the surface area formula. So, our formula becomes 120π = 2π(5)² + 2π(5)h. Let's start simplifying, step by step:

  1. Calculate the area of the two circular bases: 2π(5)² = 2π(25) = 50π.
  2. Rewrite the equation: 120Ï€ = 50Ï€ + 10Ï€h.
  3. Isolate the term with 'h': Subtract 50Ï€ from both sides. This gives us 70Ï€ = 10Ï€h.
  4. Solve for 'h': Divide both sides by 10Ï€. This leaves us with h = 7 cm.

And there you have it! The height of the cylinder is 7 cm. We have successfully navigated through the equation, step by step, ensuring that the unknown, the height, is correctly calculated. This systematic approach is vital for ensuring accuracy. By breaking down each step, we have kept things clear and easy to understand. Each of these steps plays a vital role in arriving at our solution. First, we determine the surface area of the circular ends of the cylinder. Then, we substitute the known values into the equation. Through algebraic manipulation, we isolate the unknown variable, in this case, the height. The final result represents the correct height of the cylinder. Feel proud of yourself; this is a solid math achievement!

We have successfully used the surface area formula, simplified the equation, and determined the cylinder’s height. This process not only solves the problem but also builds a solid foundation for future math problems. By approaching the problem in a systematic and organized manner, we've demonstrated how complex calculations can be broken down into manageable steps.

Verification and Conclusion

Let’s quickly check if our answer makes sense. If the height is 7 cm and the radius is 5 cm, we can calculate the surface area again to verify: SA = 2π(5)² + 2π(5)(7) = 50π + 70π = 120π cm². Awesome, it matches the original information. It is crucial to verify your answers in order to ensure you have come up with the right solution. This simple check reinforces our confidence in the result. By verifying the solution, we're ensuring that the answer is accurate and aligns with the initial conditions provided in the problem. This practice helps to improve problem-solving skills and promotes a deeper understanding of the concepts. Verifying the answer is an important step. Doing so builds confidence, reinforces understanding, and helps in the development of good problem-solving habits.

So, there you have it! We started with the surface area and radius, and through a bit of formula work, we found the height of the cylinder. Remember, geometry problems often seem daunting at first, but with a clear strategy and step-by-step approach, they're totally manageable. We've not only solved the math problem but also enhanced our understanding of cylinders and the importance of formulas. This is a clear demonstration that even the most complex problems can be simplified and understood with the right tools and approach. Keep practicing, and you'll become a geometry whiz in no time. Congratulations! You've successfully conquered another math challenge. Great job!