Calculate The Height Of A 1200 M³ Cylindrical Water Tank With A 4m Radius
Introduction
Hey guys! Ever wondered how to calculate the height of a cylindrical water tank? It's actually a pretty straightforward process, especially if you know the volume and radius. Today, we're going to dive into a real-world problem: figuring out the height of a cylindrical tank that can hold 1200 cubic meters of water, with a radius of 4 meters. This is a common problem in fields like engineering, construction, and even environmental science. Understanding how to calculate these dimensions is super important for designing efficient and safe water storage systems. We'll break down the formula, walk through the steps, and make sure you've got a solid grasp of how it all works. So, let's get started and unravel this mathematical puzzle together!
Understanding the Problem
Before we jump into the calculations, let's make sure we fully understand the problem. We have a cylindrical water tank, which, as you might know, is shaped like a cylinder – think of a can of soup or a big drum. This tank has a specific capacity, meaning it can hold a certain amount of water. In our case, it can hold 1200 cubic meters (m³) of water. This is the tank's volume. We also know that the tank has a radius of 4 meters. The radius is the distance from the center of the circular base to any point on the edge of the circle. What we need to find is the height of the tank. The height is the vertical distance from the base to the top of the cylinder. Knowing these key pieces of information helps us set up the problem correctly and choose the right formula to use. It's like having all the ingredients for a recipe – now we just need to follow the instructions to get the final result. So, let's move on to the formula we'll use to solve this.
The Formula for Volume of a Cylinder
Okay, guys, let's talk formulas! To figure out the height of our cylindrical tank, we need to use the formula for the volume of a cylinder. The formula is pretty simple and elegant: Volume (V) = π * radius² (r²) * height (h). Here, π (pi) is a mathematical constant that's approximately 3.14159. The radius (r) is the distance from the center of the circular base to the edge, and the height (h) is what we're trying to find. This formula essentially tells us that the volume of a cylinder is directly related to the area of its circular base (π * r²) and its height. Think of it like stacking circular discs on top of each other – the more discs you stack (the higher the cylinder), and the bigger the discs (the larger the radius), the more volume you have. Now that we have the formula, we can see how all the pieces fit together. We know the volume (V) and the radius (r), so we can rearrange the formula to solve for the height (h). This is where the algebra comes in, and it's actually quite fun once you get the hang of it. Let's dive into rearranging the formula in the next section!
Rearranging the Formula to Solve for Height
Alright, let's get our algebra hats on! We've got our formula for the volume of a cylinder: V = π * r² * h. But remember, we're trying to find the height (h), so we need to rearrange this formula to isolate h on one side. This is where some basic algebraic manipulation comes into play. The goal is to get h by itself, so we need to undo the operations that are being done to it. In this case, h is being multiplied by π and r². To undo multiplication, we use division. So, we'll divide both sides of the equation by π * r². This gives us: h = V / (π * r²). See how we've moved the π * r² from the right side of the equation to the denominator on the left side? Now we have a formula that directly calculates the height (h) when we know the volume (V) and the radius (r). This is a super useful skill in math and science, being able to rearrange formulas to solve for different variables. Now that we have our rearranged formula, we're ready to plug in the values we know and calculate the height of our water tank. Let's do that in the next section!
Calculation Steps
Step 1: Plug in the Values
Okay, guys, let's get to the fun part – plugging in the numbers! We know the volume (V) of the tank is 1200 cubic meters (m³) and the radius (r) is 4 meters. We also know that π (pi) is approximately 3.14159. Now, we just need to substitute these values into our rearranged formula: h = V / (π * r²). So, we get: h = 1200 / (3.14159 * 4²). This is where things start to get real. We've taken the abstract formula and made it concrete by filling in the values specific to our problem. It's like having a blank map and now we're marking the locations we need to get to. Before we can divide, we need to calculate the value of the denominator (π * r²). This means we need to square the radius (4²) and then multiply by π. Let's tackle that next.
Step 2: Calculate the Denominator
Time to crunch some numbers! We need to calculate the denominator of our equation, which is π * r². We know that r (radius) is 4 meters, so r² (radius squared) is 4 * 4 = 16 square meters. Now, we multiply this by π (approximately 3.14159): 3.14159 * 16 = 50.26544. So, the denominator is approximately 50.26544. This represents the area of the circular base of the cylinder multiplied by π. It's a crucial step in finding the height because it tells us how much area the base covers, which directly affects how tall the cylinder needs to be to hold the given volume. Now that we have the denominator, we're ready to divide the volume by this value to find the height. Let's move on to the final calculation step!
Step 3: Calculate the Height
Alright, folks, we're in the home stretch! We've got our volume (1200 m³) and our denominator (approximately 50.26544). Now, we just need to divide the volume by the denominator to find the height (h). So, h = 1200 / 50.26544. When we perform this division, we get approximately 23.87324. This means the height of the cylindrical water tank is approximately 23.87324 meters. It's always a good idea to think about the units to make sure our answer makes sense. We divided cubic meters (m³) by square meters (m²), which leaves us with meters (m), a unit of length, which is exactly what we'd expect for a height. We've successfully calculated the height of the tank! But before we celebrate, let's round our answer to a more practical number and think about the implications of our result.
Final Answer
Rounding the Result
Okay, let's talk about rounding. Our calculated height is approximately 23.87324 meters, which is pretty precise. But in the real world, measurements don't always need to be that exact. For practical purposes, we can round this number to a more manageable value. Depending on the situation, we might round to the nearest tenth, hundredth, or even whole number. For this example, let's round to the nearest tenth of a meter. Looking at our number, 23.87324, the digit in the tenths place is 8, and the digit to the right of it is 7, which is 5 or greater. So, we round up. This means our height, rounded to the nearest tenth of a meter, is 23.9 meters. Rounding makes our answer easier to work with and communicate, especially in practical applications like construction or engineering. Now that we have our final, rounded answer, let's state it clearly and consider what it means in the context of our problem.
The Height of the Tank
So, guys, after all our calculations, we've arrived at the final answer! The height of the cylindrical water tank is approximately 23.9 meters. That's pretty tall, right? This means that to hold 1200 cubic meters of water with a radius of 4 meters, the tank needs to be almost 24 meters high. This result is super important for anyone designing or building this kind of tank. Knowing the height, along with the radius, allows engineers to ensure the structural integrity of the tank and make sure it can safely hold the intended amount of water. It also helps with planning the overall layout and placement of the tank in a given space. We've not only solved a mathematical problem, but we've also gained insight into a real-world application. Math is all around us, and understanding these concepts can be incredibly useful. Now, let's wrap things up with a quick summary of what we've learned and why it's important.
Conclusion
Summary of Steps
Hey guys, let's recap what we've learned today! We set out to find the height of a cylindrical water tank given its volume (1200 m³) and radius (4 m). We started by understanding the problem and identifying the key information. Then, we introduced the formula for the volume of a cylinder: V = π * r² * h. We realized that we needed to find the height (h), so we rearranged the formula to solve for h: h = V / (π * r²). Next, we plugged in the values we knew into the rearranged formula. We calculated the denominator (π * r²) and then divided the volume by the denominator to find the height. We got an initial answer of approximately 23.87324 meters. Finally, we rounded this result to a more practical value of 23.9 meters. We not only solved the problem but also discussed the importance of each step and how it relates to real-world applications. Understanding these steps is crucial for tackling similar problems and for applying mathematical concepts in practical situations. But why is this important beyond just solving a math problem?
Importance of the Calculation
So, why is calculating the height of a cylindrical tank important, you might ask? Well, this kind of calculation is fundamental in many fields. In engineering, it's essential for designing storage tanks for water, fuel, chemicals, and other liquids. Knowing the correct dimensions ensures that the tank can hold the required volume safely and efficiently. In construction, understanding these calculations is crucial for building foundations and structures that can support the weight of the tank and its contents. In environmental science, these calculations can help in designing water treatment facilities and storage systems for managing water resources. The applications are vast and varied. Beyond the specific example of a water tank, the principles we've learned – understanding formulas, rearranging equations, and applying them to real-world problems – are transferable to countless other situations. Math isn't just about numbers; it's about problem-solving and critical thinking. And that's a skill that's valuable in any field. So, the next time you see a cylindrical tank, you'll know the math behind it and appreciate the importance of these calculations. Keep exploring, keep learning, and keep applying these concepts to the world around you!
