Calculating Sphere Volume When Radius Is 3 Inches

Hey everyone! Today, we're diving into the world of geometry to tackle a common question involving spheres. Specifically, we're going to figure out how to calculate the volume of a sphere when we know its radius. This is a fundamental concept in mathematics, and mastering it will definitely help you in various real-world applications and further studies.

Understanding the Question The Radius of a Sphere is 3 Inches. Which Represents the Volume of the Sphere?

Let's break down the problem. We're given a sphere, which is a perfectly round three-dimensional object, like a ball. The key piece of information we have is the radius, which is the distance from the center of the sphere to any point on its surface. In this case, the radius is 3 inches. Our mission is to determine the volume of this sphere. The volume, in simple terms, is the amount of space the sphere occupies, and it's measured in cubic inches because we're dealing with a three-dimensional object. The options provided are:

• A. 12 cubic inches
• B. 36 cubic inches
• C. 64 cubic inches
• D. 81 cubic inches

To solve this, we need to dust off our knowledge of the formula for the volume of a sphere. Don't worry if you don't remember it off the top of your head; we're going to walk through it together. This formula is a cornerstone in geometry, allowing us to move from a simple measurement like the radius to a comprehensive understanding of the sphere's size. So, let’s roll up our sleeves and get into the heart of the calculation. We’ll start by revisiting the formula itself and then apply it step-by-step to find the correct answer. Stick with me, and you'll see that calculating the volume of a sphere is not as daunting as it might seem!

The Formula for the Volume of a Sphere

Alright, let's talk formulas! The volume of a sphere is calculated using a specific formula that relates the volume to the sphere's radius. This formula is a crucial tool in geometry, and it's expressed as:

V = (4/3)πr³

Where:

• V stands for the volume of the sphere (what we're trying to find).
• (4/3) is a constant fraction.
• π (pi) is a mathematical constant approximately equal to 3.14159.
• r represents the radius of the sphere.
• The r³ means we need to cube the radius (multiply it by itself three times).

This formula might look a bit intimidating at first, but it's actually quite straightforward once you understand each component. The constant (4/3) and π are always the same, so the only variable that changes is the radius. This means that if we know the radius, we can plug it into the formula and calculate the volume. Understanding why this formula works involves delving into calculus, but for our purposes, we can simply accept it as a given rule. Think of it as a recipe: if you follow the ingredients (the formula) and the instructions (the steps), you'll get the desired result (the volume).

Now, let’s take a closer look at each part of the formula and what it signifies. The (4/3) factor is a result of the mathematical derivation of the sphere's volume, a concept rooted in integral calculus. Pi (π) is that famous number that shows up in all sorts of circle and sphere calculations, representing the ratio of a circle's circumference to its diameter. And the r³, cubing the radius, is what makes the volume calculation three-dimensional, reflecting the sphere's nature as a 3D object. With this formula in our toolkit, we’re now ready to apply it to our specific problem, where the radius is given as 3 inches. So, let’s move on to the next step: plugging in the values and crunching the numbers!

Applying the Formula to Our Problem

Now comes the fun part: let's use the formula we just discussed to solve our problem. We know the radius (r) of the sphere is 3 inches. So, we're going to substitute this value into our volume formula:

V = (4/3)πr³

Replace r with 3:

V = (4/3)π(3)³

First, we need to calculate 3 cubed (3³), which means 3 * 3 * 3. This equals 27. So our equation now looks like this:

V = (4/3)π(27)

Next, we can multiply (4/3) by 27. To do this, you can think of it as (4 * 27) / 3. 4 multiplied by 27 is 108, and then we divide that by 3, which gives us 36. Now our equation is:

V = 36π

This means the volume is 36 times π. Since π is approximately 3.14159, we can multiply 36 by this value to get a more precise answer. However, if we look at our answer choices, they are in terms of whole numbers, so we might not need to do the exact multiplication just yet. Let’s hold off on that for a moment and see if we can identify the correct answer without it.

By substituting the radius into the formula and simplifying, we've arrived at V = 36π. This is a significant step forward. We've reduced the problem to a simple expression involving π, and we're now in a good position to compare our result with the given options. The key here was to follow the order of operations (cubing the radius first, then multiplying by the constants) and to keep track of our calculations. Now, the final step is to match our calculated volume with one of the provided answers. This will not only give us the solution to the problem but also reinforce our understanding of how the formula works in practice. So, let's move on to the final phase: comparing our result and choosing the correct answer!

Comparing the Result with the Options

Okay, guys, we've arrived at the equation V = 36π. Now, let's compare this with the options given in the question:

• A. 12 cubic inches
• B. 36 cubic inches
• C. 64 cubic inches
• D. 81 cubic inches

Notice that our result, 36π, includes π. This means the actual numerical value of the volume will be 36 multiplied by approximately 3.14159, which will be a number a little bit more than 36 multiplied by 3. Looking at the options, we need to figure out which one is closest to 36π.

If we think about it, 36π is definitely going to be larger than 36 because we're multiplying 36 by a number greater than 1 (π is about 3.14). So, option B, which is exactly 36 cubic inches, is not the correct answer. Option A, 12 cubic inches, is much too small. Now we are left with option C, 64 cubic inches and option D, 81 cubic inches. If we approximate π as 3, then 36π is approximately 36 * 3 = 108, both options C and D don't match, so, we need to find the best option among these, so we must return to the exact form without π. We have deduced V=36π, there may be some error in the question options.

Even though we've pinpointed that V=36π, the provided options seem to deviate from this correct form. This situation underscores the importance of understanding the underlying principles and calculations, even when faced with discrepancies. Our step-by-step approach, from recalling the formula to substituting values and simplifying, has led us to a clear and accurate result. In real-world scenarios, this level of precision and understanding is invaluable for problem-solving and decision-making. So, while the exact match may not be present in the given choices, our journey through the calculation has equipped us with the knowledge and skills to confidently tackle similar challenges in the future. Remember, the goal isn't just about finding the right answer in a multiple-choice question; it's about mastering the process and being able to apply it in various contexts.

Conclusion

In conclusion, we tackled the problem of finding the volume of a sphere with a radius of 3 inches. We revisited the formula for the volume of a sphere, V = (4/3)πr³, and carefully substituted the given radius into the formula. Through step-by-step calculations, we arrived at the result V = 36π cubic inches. However, when comparing our result with the provided options, we encountered a discrepancy, as none of the options perfectly matched our calculated volume.

Despite this, the process of solving the problem was a valuable exercise in understanding and applying geometric formulas. We reinforced the importance of accurately recalling and using formulas, performing calculations systematically, and interpreting results in the context of the problem. Even when faced with unexpected outcomes, such as mismatched answer choices, the knowledge and skills gained through the problem-solving process remain invaluable.

So, while we didn't find an exact match in the given options, we successfully navigated the problem, demonstrating a solid understanding of how to calculate the volume of a sphere. This experience highlights that in mathematics, the journey of problem-solving is just as important as the destination. Keep practicing, keep exploring, and you'll continue to strengthen your mathematical abilities!