Calculate Sphere Radius From Surface Area 320 Sq Cm

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Hey guys! Today, we're diving into a fun little math problem involving spheres. Specifically, we're going to figure out how to calculate the radius of a sphere when we know its surface area. Our challenge is this: The surface area of a sphere is 320 square centimeters. What is the radius of the sphere, rounded to 2 decimal places? Sounds interesting, right? Let's break it down step by step.

Understanding the Formula

Before we jump into the calculations, let's quickly recap the formula for the surface area of a sphere. This is our trusty tool that will guide us to the answer. The formula is:

Surface Area (SA) = 4πr²

Where:

  • SA is the surface area of the sphere
  • Ï€ (pi) is a mathematical constant approximately equal to 3.14159
  • r is the radius of the sphere (what we're trying to find!)

Now that we've got the formula in our toolkit, let's understand why this formula is so important and how it connects the surface area and the radius. Imagine peeling an orange – the peel represents the surface area. The radius, on the other hand, is the distance from the center of the orange to any point on its surface. The formula tells us that the surface area is directly related to the square of the radius. This means if you double the radius, the surface area increases by a factor of four! This relationship is crucial in many real-world applications, from designing spherical tanks to calculating the size of planets.

Our main keyword here is calculating the radius of a sphere, so let's keep that in mind as we move forward. The formula itself is a powerhouse of mathematical elegance, neatly packing a complex geometric relationship into a simple equation. It allows us to move between surface area and radius seamlessly, which is exactly what we need to do in our problem. Remember, we know the surface area (320 square centimeters), and we're hunting for the radius. To find it, we'll need to rearrange the formula and use some algebraic magic. But before we do that, let's take a moment to appreciate the beauty and power of this formula – it's the key to unlocking the mystery of our sphere!

Rearranging the Formula to Solve for the Radius

Okay, now for the fun part – let's rearrange the formula to isolate the radius (r). This is where our algebra skills come into play. We want to get 'r' all by itself on one side of the equation. Here's how we do it:

  1. Start with the formula: SA = 4πr²
  2. Divide both sides by 4π: SA / (4π) = r²
  3. Take the square root of both sides: √(SA / (4π)) = r

Voila! We've successfully rearranged the formula. Now we have an equation that directly tells us the radius in terms of the surface area:

r = √(SA / (4π))

This rearranged formula is our golden ticket to solving the problem. It's like having a secret decoder ring that translates surface area into radius. But let's not just rush into plugging in numbers just yet. Let's think about why this rearrangement works. By dividing by 4π, we're essentially undoing the multiplication that was happening to r². And by taking the square root, we're undoing the squaring operation. This is a classic example of using inverse operations in algebra to solve for an unknown variable. Understanding the why behind the formula is just as important as knowing the formula itself. It helps us build a deeper connection with the math and makes it easier to remember and apply in different situations.

Our focus remains on calculating the radius of a sphere, and this rearranged formula is the key to unlocking that calculation. It allows us to take a known surface area and directly compute the radius. Now we're ready to put this formula to work with our specific problem. So, let's move on to the next step: plugging in the given surface area and crunching the numbers!

Plugging in the Values and Calculating the Radius

Alright, with our rearranged formula in hand, we're ready to plug in the surface area and get our hands on that radius! Remember, we're given that the surface area (SA) is 320 square centimeters. So, let's substitute that into our formula:

r = √(320 / (4π))

Now, it's just a matter of crunching the numbers. First, let's simplify the expression inside the square root:

r = √(320 / (4 * 3.14159)) (approximating π as 3.14159)

r = √(320 / 12.56636)

r = √(25.46479)

Now, we take the square root:

r ≈ 5.04627 centimeters

But wait! We're not quite done yet. The problem asks us to round our answer to 2 decimal places. So, let's do that:

r ≈ 5.05 centimeters

And there we have it! We've successfully calculated the radius of the sphere. It's pretty cool how a single formula, combined with a bit of algebra and arithmetic, can give us such a precise answer. The process of calculating the radius of a sphere involves more than just plugging numbers into a formula. It's about understanding the relationship between the surface area and the radius, knowing how to manipulate equations, and paying attention to the details (like rounding!). Each step in the calculation is important, and double-checking our work ensures we arrive at the correct answer. Now that we've got our unrounded answer, let's make sure we present it in the format requested by the problem. Rounding to two decimal places is a common practice in many scientific and engineering applications, so it's a good habit to develop. So, our final answer, rounded to two decimal places, is approximately 5.05 centimeters.

Final Answer and Conclusion

So, after all that math magic, we've arrived at our final answer! The radius of the sphere with a surface area of 320 square centimeters, rounded to 2 decimal places, is approximately 5.05 centimeters.

Answer: 5.05 cm

Great job, everyone! We took a problem, broke it down into smaller, manageable steps, and conquered it. We revisited the formula for the surface area of a sphere, rearranged it to solve for the radius, plugged in the given values, and performed the calculations. We even remembered to round our answer to the specified decimal places. This is a perfect example of how math can be used to solve real-world problems. Whether you're designing a spherical container, calculating the size of a balloon, or even studying the planets, understanding the relationship between surface area and radius is essential.

The key takeaway here isn't just the numerical answer, but the process we followed to get there. We started with a problem, identified the relevant formula, manipulated it to suit our needs, and then executed the calculations with precision. This problem-solving approach is applicable not just in math, but in many other areas of life. And remember, calculating the radius of a sphere is just one example of the power and versatility of mathematics. There are countless other geometric shapes and formulas waiting to be explored. So, keep practicing, keep asking questions, and keep exploring the wonderful world of math! You might be surprised at what you discover.