Finding The Radius Of A Barium Cylinder: Equations & Solutions

Hey math enthusiasts! Ever wondered how to figure out the radius of a cylinder made of barium, knowing its mass and a specific formula? Well, buckle up, because we're diving into that very topic! We'll explore how to set up the right equations and find the solution. Let's break it down, step by step, making sure it's super clear and easy to follow. Get ready to flex those math muscles!

Understanding the Problem: The Barium Cylinder's Mass

Alright, so here's the deal: We've got a cylinder made of barium, and we know its height is 2 inches. The mass of this cylinder (let's call it M) depends on its radius (r), and we've got a handy-dandy function to describe this relationship: $M(r) = 7.188 imes imes ext{pi} imes r^2$. This formula tells us that if we know the radius, we can calculate the mass. The real question is: How do we work backward? If we know the mass, how do we find the radius? That's where the system of equations comes into play. It's like a secret code we use to unlock the mystery of the cylinder's dimensions. Think of the function as a recipe: It tells you how to make mass from the radius. But, we want to know what ingredient (the radius) to use if we have the final dish (the mass). That's what we are going to find out. This is all about applying our knowledge to a real-world (or at least, a hypothetical world) scenario, making math fun and practical. The beauty of math is that it helps us understand and solve problems like this one.

Breaking Down the Function

The function $M(r) = 7.188 imes ext{pi} imes r^2$ is our starting point. Let's break it down a bit. M(r) represents the mass of the cylinder, and it's a function of r (the radius). The number 7.188 is likely related to the density of barium (how much mass is packed into a given volume). The $ ext{pi}$ (approximately 3.14159) comes into play because we are dealing with a cylinder, which has a circular base. The r² means the radius is squared, and that relates to the area of the circle that makes up the base of the cylinder. It will be helpful to remember this: The mass is directly influenced by the radius. A larger radius will lead to a bigger mass, and this relationship is key to understanding how to solve the problem. Every part of this formula is important, and understanding each part gives you the power to find your desired radius. If you had to make a cylinder with a particular mass, the information presented would allow you to calculate the radius needed. This is the goal we are aiming for.

The Importance of the Radius

The radius is a fundamental element of the cylinder's structure. If we change it, everything else changes. Think of it like a building's foundation: a wider foundation will allow the building to be higher without falling, so the radius of the barium cylinder is a critical number. It dictates the cylinder's base size and, consequently, its volume (and thus its mass). Knowing the radius is essential for manufacturing or working with barium cylinders in engineering, physics, and even chemistry. Imagine you're designing something, and you need a cylinder of a specific mass. Without knowing the radius, you can't proceed. That's why solving for the radius, given the mass, is a super useful skill. It's like having a key that unlocks a whole range of possibilities. Every detail matters, so don't overlook it; now we can dive right into finding out how to find the radius.

Setting Up the System of Equations

Okay, now let's get down to the nitty-gritty of setting up our system of equations. Our goal is to find the radius (r) when we know the mass (M). The key here is to have a specific mass value to work with. Let's say, for example, that the mass of our barium cylinder is 100 grams. Now, we can create our equation. This is not just about writing down formulas; it's about translating a real-world problem into a mathematical language we can understand and solve. It's a method to turn words and concepts into clear steps. The system of equations is the tool that enables us to take the information given (the mass) and use the equation given ($M(r)=7.188 imes ext{pi} imes r^2$) to find the unknown (the radius). It’s like a puzzle: each piece of the equation contributes to the solution.

The Basic Equation

We already know our main equation: $M(r) = 7.188 imes ext{pi} imes r^2$. Since we know the mass, let's substitute a value for M(r). In our example, we are using 100 grams. This gives us: $100 = 7.188 imes ext{pi} imes r^2$. That's the equation we will solve. It's all about figuring out the value of r that makes this equation true. Now, this single equation is our system in this particular scenario. There is no need for multiple equations because we have one equation with one unknown variable, which is the radius. As long as you know the mass, we have the correct tools to get a solution.

Understanding the Goal

Our primary objective is isolating r (the radius). We'll do this using a series of algebraic manipulations, which will involve dividing, and square roots. The goal is to rearrange the equation so that r stands alone on one side, and on the other side, we'll have a number. That number will be the radius of our barium cylinder. It's like peeling back layers of the equation until you find the hidden treasure (the radius). Keep in mind the equation tells you the mass of the cylinder, once the radius is known, so the goal is to reverse the process.

Solving for the Radius: Step-by-Step

Alright, time to get our hands dirty and actually solve for r! This part is about applying the algebraic skills we have learned. We will use those skills to isolate the radius in our equation. We will be using the equation $100 = 7.188 imes ext{pi} imes r^2$. Follow these steps to get the solution. Remember, that math can be as fun as you want it to be.

Step 1: Divide by the Constants

First, we want to isolate r². To do this, we'll divide both sides of the equation by $7.188 imes ext{pi}$. This gives us: $r^2 = 100 / (7.188 imes ext{pi})$. This step simplifies the equation, getting us closer to solving for r. You'll find yourself using these basic algebraic manipulations all the time in math and science, so this is a great exercise. The important thing to keep in mind is that, whatever operation you perform on one side of the equation, you must perform on the other to keep things balanced. Otherwise, the equation becomes false.

Step 2: Calculate the Constant Term

Next, calculate the value of $100 / (7.188 imes ext{pi})$. This gets us a numerical value for $r^2$. Using a calculator, $7.188 imes ext{pi} imes 22.60$, so that will give us $r^2 ext{ approximately } 4.42$. Doing calculations is a critical part of solving any math problem. We now know that $r^2$ is equal to 4.42. These constant calculations will help you get closer to your solution. Now that we have the square of the radius, we can determine the radius itself.

Step 3: Take the Square Root

Finally, to solve for r, we take the square root of both sides of the equation: $r = ext{sqrt}(4.42)$. Using a calculator, we find that the square root of 4.42 is approximately 2.1. Therefore, r is approximately 2.1 inches. The last step gives us our final answer: the radius of our barium cylinder is about 2.1 inches. Remember to always include the units (inches in this case) to make sure your answer makes sense. Taking the square root is the final step in determining the radius. So now you know how to do it!

Conclusion: Finding the Radius Made Easy!

There you have it! We've successfully calculated the radius of a barium cylinder given its mass using a system of equations. We started with the basic formula, substituted our known mass, and then used some simple algebra to solve for r. The process wasn't too difficult, right? You should be proud of yourself for solving a real-world problem using math. Remember, the concepts we covered here can be applied to many different scenarios. Math becomes powerful when it is applicable to real-life situations. The system of equations is a fundamental concept in mathematics that has endless applications. Hopefully, this gave you a better understanding of how to find the radius of a cylinder made of barium, but also how to solve the problem by creating the right equation. You now know everything you need to know.

Key Takeaways

• We used the formula $M(r) = 7.188 imes ext{pi} imes r^2$ to represent the mass of the cylinder.
• We set up the equation $100 = 7.188 imes ext{pi} imes r^2$ where 100 is the mass of our barium cylinder.
• We found the radius, r, by dividing and then taking the square root. These steps gave us a value for r of approximately 2.1 inches.

Now go forth and apply your new knowledge! Keep practicing, and you'll become a math whiz in no time. If you have any questions, feel free to ask! Math can be super exciting once you get the hang of it, and we are happy to help in any way we can!