Finding The Radius Of A Sphere Given Its Volume 81π Cubic Centimeters

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In the realm of geometry, the sphere stands as a fundamental three-dimensional object, characterized by its perfect symmetry and constant curvature. Understanding the properties of spheres, including their volume and surface area, is crucial in various scientific and engineering applications. In this comprehensive guide, we will delve into the formula for the volume of a sphere and apply it to solve a specific problem: determining the equation that can be used to find the radius of a sphere given its volume. Specifically, we will address the problem where the volume of a sphere is given as 81π cubic centimeters.

The volume of a sphere, which represents the amount of space it occupies, is mathematically defined by a simple yet elegant formula: $V = \frac{4}{3}πr^3$, where V denotes the volume and r represents the radius of the sphere. This formula is derived using calculus and involves integrating the area of a circle over the sphere's three-dimensional space. The constant π (pi) is an irrational number approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter. Understanding this formula is paramount to solving problems related to spheres.

In essence, the formula tells us that the volume of a sphere is directly proportional to the cube of its radius. This means that a small change in the radius can lead to a significant change in the volume. For example, if you double the radius of a sphere, its volume increases by a factor of eight (2 cubed). This relationship is crucial in many real-world applications, such as designing spherical tanks or calculating the amount of material needed to create spherical objects.

Moreover, the formula highlights the importance of π in spherical geometry. This constant appears ubiquitously in formulas involving circles and spheres, underscoring the fundamental connection between these geometric shapes. The presence of π in the volume formula also indicates that the volume is inherently related to the circular cross-sections of the sphere. Imagine slicing the sphere into infinitesimally thin circular disks; the sum of the volumes of these disks, integrated over the sphere's radius, yields the total volume formula.

Now, let's consider the problem at hand. We are given that the volume of a sphere is 81π cubic centimeters, and our objective is to identify the equation that can be used to find the radius of this sphere. This is a classic problem in geometry that requires us to apply our understanding of the volume formula and algebraic manipulation.

To solve this, we need to set up an equation that equates the given volume with the volume formula. This will allow us to isolate the radius r and solve for its value. The key here is to correctly substitute the given volume into the formula and then rearrange the equation to match one of the provided options. This process not only tests our knowledge of the formula but also our ability to apply it in a practical context.

  1. Start with the volume formula: The fundamental equation for the volume of a sphere is $V = \frac{4}{3}πr^3$. This is our starting point, and it encapsulates the mathematical relationship between the volume and the radius.

  2. Substitute the given volume: We are given that the volume V is 81π cubic centimeters. Substitute this value into the formula: $81π = \frac{4}{3}πr^3$. This step is crucial as it sets up the equation we need to solve for the radius. By replacing V with its known value, we transform the general formula into a specific equation for our problem.

  3. Compare with the given options: Now, we need to compare the equation we have obtained with the options provided in the problem. This involves careful observation and pattern matching. Our goal is to identify which of the given equations matches the form we have derived.

  4. Correct Equation Identification: The equation $81π = \frac{4}{3}πr^3$ matches one of the provided options, thus it is the correct equation to use to find the radius of the sphere.

To fully understand the solution, it is important to examine why the other provided options are incorrect. This not only reinforces our understanding of the volume formula but also sharpens our analytical skills in identifying errors.

  • Option 1: 81π = 4πr²: This equation resembles the formula for the surface area of a sphere, which is 4πr², not the volume. Using this equation would lead to an incorrect value for the radius. The presence of r² instead of r³ clearly indicates that this formula is inappropriate for calculating volume.

  • Option 2: 81π = 4πr³: This equation is similar to the correct volume formula but lacks the crucial fraction \frac{4}{3}. This omission significantly alters the relationship between the volume and the radius, leading to an incorrect result. The coefficient 4π is part of the complete volume formula, but without the \frac{1}{3} factor, it does not accurately represent the volume of a sphere.

While the problem asks only for the equation, let's go a step further and solve for the radius r. This will provide a complete understanding of the problem and demonstrate how the equation is used in practice.

  1. Isolate r³: To solve for r, we need to isolate r³ on one side of the equation. Start by multiplying both sides of the equation $81π = \frac4}{3}πr^3$ by \frac{3}{4} $(\frac{3{4})81π = πr^3$ This simplifies to $\frac{243}{4}π = πr^3$

  2. Divide by π: Next, divide both sides by π to eliminate it from the equation: $\frac{243}{4} = r^3$

  3. Take the cube root: To find r, take the cube root of both sides: $r = \sqrt[3]{\frac{243}{4}}$ This gives us the exact value of the radius.

  4. Simplify (Optional): We can simplify the expression further, if needed, to obtain a numerical approximation or a simplified radical form. However, for the purpose of this problem, leaving the answer in exact form is perfectly acceptable.

The ability to calculate the volume and radius of spheres has numerous practical applications in various fields:

  • Engineering: Engineers use these calculations to design spherical tanks for storing liquids or gases, ensuring they can hold the required volume. For instance, in the design of pressure vessels or storage tanks for natural gas, accurate volume calculations are crucial for safety and efficiency.

  • Physics: Physicists use the volume of spheres to calculate the density and mass of spherical objects, such as planets or stars. Understanding the volume and mass relationships is fundamental in astrophysics and cosmology.

  • Medicine: In medical imaging, the volume of spherical tumors or cysts can be calculated to monitor their growth or shrinkage over time. This is a vital tool in cancer treatment and diagnostics.

  • Manufacturing: Manufacturers use these calculations to determine the amount of material needed to produce spherical products, such as ball bearings or spherical containers. Accurate material estimation helps in cost management and production planning.

In summary, the equation that can be used to find the radius of a sphere with a volume of 81π cubic centimeters is $81π = \frac{4}{3}πr^3$. This problem demonstrates the practical application of the volume formula and the importance of understanding the relationship between a sphere's volume and its radius. By following a step-by-step approach, we can confidently solve such problems and appreciate the significance of spherical geometry in various real-world scenarios. From engineering design to medical diagnostics, the principles of spherical geometry play a crucial role in our understanding and manipulation of the physical world.