Sphere Volume Equation Given Cylinder Volume

Understanding the relationship between geometric shapes is a cornerstone of mathematics. In this article, we'll delve into a problem involving a sphere and a cylinder with identical radii and heights. Given the cylinder's volume, our task is to pinpoint the equation that accurately represents the sphere's volume. This exploration will not only reinforce your understanding of volume formulas but also highlight the elegant connections between different geometric figures.

Problem Statement: Decoding the Volumes

Let's restate the problem to ensure clarity. We are presented with a sphere and a cylinder that share the same radius and height. We know that the cylinder's volume is $27 \pi \text{ ft}^3$. The central question we aim to answer is: Which equation correctly expresses the volume of the sphere under these conditions?

To solve this, we must consider the volume formulas for both the cylinder and the sphere. The formula for the volume of a cylinder is given by:

$$V_{\text{cylinder}} = \pi r^2 h$$

where:

• V

  _cylinder is the volume of the cylinder

• r is the radius of the cylinder

• h is the height of the cylinder

The formula for the volume of a sphere is:

$$V_{\text{sphere}} = \frac{4}{3} \pi r^3$$

where:

• V

  _sphere is the volume of the sphere

• r is the radius of the sphere

The key to unlocking this problem lies in recognizing the relationship between the cylinder's height and the sphere's radius. Since the sphere and cylinder have the same radius and height, and the height of the cylinder is equivalent to the diameter of the sphere (2r), we can establish a direct link between the two volumes.

Deconstructing the Cylinder's Volume

Our starting point is the cylinder's volume, which is given as $27 \pi \text{ ft}^3$. We can use this information in conjunction with the cylinder volume formula to glean insights about the radius and height. Substituting the given volume into the formula, we get:

$$27 \pi = \pi r^2 h$$

Notice that $\pi$ appears on both sides of the equation. We can divide both sides by $\pi$ to simplify:

$$27 = r^2 h$$

This equation reveals a crucial relationship between the radius squared and the height of the cylinder. Remember, the height of the cylinder is equal to the diameter of the sphere, which is 2r. We can substitute h with 2r in our equation:

$$27 = r^2 (2r)$$

Simplifying further:

$$27 = 2r^3$$

Now, we can isolate $r^3$ by dividing both sides by 2:

$$r^3 = \frac{27}{2}$$

This value of $r^3$ will be pivotal in determining the sphere's volume.

Constructing the Sphere's Volume Equation

With $r^3$ in hand, we can now turn our attention to the sphere's volume formula:

$$V_{\text{sphere}} = \frac{4}{3} \pi r^3$$

The brilliance of this formula is how it directly incorporates $r^3$, a value we've already calculated. Substituting $r^3 = \frac{27}{2}$ into the sphere's volume formula, we get:

$$V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{27}{2} \right)$$

This equation represents the volume of the sphere in terms of known quantities. We can further simplify it to match one of the provided options.

Let's break down the simplification step-by-step:

1. Multiply the fractions: $V_{\text{sphere}} = \frac{4 \times 27}{3 \times 2} \pi$
2. Simplify the numerator and denominator: $V_{\text{sphere}} = \frac{108}{6} \pi$
3. Divide 108 by 6: $V_{\text{sphere}} = 18 \pi$

While this simplified result is valuable, the problem asks for the equation that gives the volume, not the simplified volume itself. We need to trace back through our steps to identify the equation that matches one of the provided options. Going back to the substitution step:

$$V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{27}{2} \right)$$

We can rearrange this equation to resemble the given options more closely. Notice that we can rewrite $\frac{27}{2}$ as $\frac{1}{2} \times 27$. Substituting this back into the equation:

$$V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{1}{2} \times 27 \right)$$

Rearranging the terms, we get:

$$V_{\text{sphere}} = \frac{4}{3} \times \frac{1}{2} (27 \pi)$$

Multiplying the fractions $\frac{4}{3}$ and $\frac{1}{2}$, we get $\frac{4}{6}$, which simplifies to $\frac{2}{3}$. Therefore, the equation becomes:

$$V_{\text{sphere}} = \frac{2}{3} (27 \pi)$$

This equation perfectly matches one of the given options.

Analyzing Incorrect Options: Why They Don't Fit

To solidify our understanding, let's briefly examine why the other provided equation is incorrect. The other option is:

$$V = \frac{4}{6} r \times 27$$

This equation deviates significantly from the correct approach. It incorrectly introduces the radius 'r' as a multiplicative factor outside the context of the volume formula. The volume of the sphere depends on $r^3$, not 'r' itself. Furthermore, this equation lacks the crucial $\pi$ term, which is fundamental to any volume calculation involving circles or spheres. Therefore, this equation is fundamentally flawed and cannot represent the sphere's volume in this scenario.

Conclusion: The Equation Unveiled

In conclusion, the equation that accurately gives the volume of the sphere, given the cylinder's volume of $27 \pi \text{ ft}^3$ and the shared radius and height, is:

$$V_{\text{sphere}} = \frac{2}{3} (27 \pi)$$

This solution highlights the importance of understanding and applying the correct volume formulas, recognizing the relationships between geometric dimensions, and performing algebraic manipulations to arrive at the desired equation. By dissecting the problem step-by-step and understanding the underlying principles, we've successfully navigated the complexities of volume calculations and revealed the equation that governs the sphere's volume in this specific context. This exercise not only strengthens your problem-solving skills but also deepens your appreciation for the interconnectedness of mathematical concepts in geometry.