Volleyball In A Cylinder Calculating Empty Space

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Introduction

In this article, we will solve a classic geometry problem that involves calculating the empty space within a cylinder that contains a volleyball. This is a common type of problem in mathematics that combines the concepts of volume calculation for both spheres and cylinders. By understanding the formulas for these shapes and applying some basic arithmetic, we can determine the volume of the space not occupied by the volleyball. This exercise not only reinforces our understanding of geometric formulas but also highlights the practical applications of mathematics in real-world scenarios.

Our specific challenge is to find the empty space inside a cylinder that perfectly fits a volleyball. The volleyball has a radius of 13 inches, which is also the radius of the cylinder. We will use 3.14 as an approximation for $\pi$ and round our final answer to the nearest whole number. This step-by-step solution will demonstrate how to break down the problem, apply the relevant formulas, and arrive at the correct answer. So, let’s dive in and explore the fascinating world of geometric calculations!

Problem Statement

Let's define the problem more precisely. We have a volleyball with a radius of 13 inches. This volleyball fits perfectly inside a cylinder, meaning the cylinder has the same radius as the volleyball (13 inches) and its height is equal to the diameter of the volleyball. Our goal is to calculate the volume of the empty space within the cylinder—that is, the space not occupied by the volleyball. To do this, we will need to use the formulas for the volume of a sphere (the volleyball) and the volume of a cylinder. By subtracting the volume of the sphere from the volume of the cylinder, we will find the empty space. Remember, we are using 3.14 as an approximation for $\pi$ and will round our final answer to the nearest whole number.

Breaking Down the Problem

To solve this problem efficiently, we need to break it down into smaller, manageable steps:

  1. Identify the Given Information: List all the known values, such as the radius of the volleyball (and cylinder) and the approximation for $\pi$.
  2. Determine the Required Formulas: Recall the formulas for the volume of a sphere and the volume of a cylinder. These are essential for our calculations.
  3. Calculate the Volume of the Volleyball (Sphere): Use the formula for the volume of a sphere and the given radius to find the volume of the volleyball.
  4. Determine the Height of the Cylinder: Since the volleyball fits perfectly inside the cylinder, the height of the cylinder is equal to the diameter of the volleyball.
  5. Calculate the Volume of the Cylinder: Use the formula for the volume of a cylinder, along with the radius and height, to find the volume of the cylinder.
  6. Calculate the Empty Space: Subtract the volume of the volleyball from the volume of the cylinder to find the volume of the empty space.
  7. Round the Answer: Round the final answer to the nearest whole number as instructed.

By following these steps, we can systematically solve the problem and arrive at the correct solution. Each step is crucial, and a clear understanding of the process will help in tackling similar problems in the future.

Formulas Required

To solve this problem, we need to recall two fundamental formulas from geometry: the volume of a sphere and the volume of a cylinder.

Volume of a Sphere

The volume of a sphere is given by the formula:

Vsphere=43Ï€r3V_\text{sphere} = \frac{4}{3} \pi r^3

Where:

  • V_\text{sphere}$ represents the volume of the sphere.

  • \pi$ (pi) is a mathematical constant, approximately equal to 3.14 in our case.

  • r$ is the radius of the sphere.

This formula tells us that the volume of a sphere is directly proportional to the cube of its radius. This means that even a small increase in the radius can result in a significant increase in the volume.

Volume of a Cylinder

The volume of a cylinder is given by the formula:

Vcylinder=Ï€r2hV_\text{cylinder} = \pi r^2 h

Where:

  • V_\text{cylinder}$ represents the volume of the cylinder.

  • \pi$ (pi) is the same mathematical constant, approximately 3.14.

  • r$ is the radius of the base of the cylinder.

  • h$ is the height of the cylinder.

This formula shows that the volume of a cylinder is the product of the area of its base (a circle with area $\pi r^2$) and its height. Understanding these formulas is crucial for solving the problem at hand. We will use them to calculate the volumes of the volleyball and the cylinder, respectively, and then find the difference to determine the empty space.

Step-by-Step Solution

Now that we have identified the problem and the necessary formulas, let's proceed with the step-by-step solution.

1. Identify the Given Information

We are given the following information:

  • Radius of the volleyball (sphere), $r = 13$ inches
  • Value of $\pi$ to use, $\pi = 3.14$

2. Calculate the Volume of the Volleyball (Sphere)

Using the formula for the volume of a sphere, $V_\text{sphere} = \frac{4}{3} \pi r^3$, we can substitute the given values:

Vsphere=43×3.14×(13)3V_\text{sphere} = \frac{4}{3} \times 3.14 \times (13)^3

Vsphere=43×3.14×2197V_\text{sphere} = \frac{4}{3} \times 3.14 \times 2197

Vsphere=4×3.14×21973V_\text{sphere} = \frac{4 \times 3.14 \times 2197}{3}

Vsphere=27638.483V_\text{sphere} = \frac{27638.48}{3}

Vsphere=9212.826666...V_\text{sphere} = 9212.826666...

So, the volume of the volleyball is approximately 9212.83 cubic inches.

3. Determine the Height of the Cylinder

Since the volleyball fits perfectly inside the cylinder, the height of the cylinder is equal to the diameter of the volleyball. The diameter is twice the radius:

h=2×rh = 2 \times r

h=2×13h = 2 \times 13

h = 26$ inches Thus, the height of the cylinder is 26 inches. ### 4. Calculate the Volume of the Cylinder Using the formula for the volume of a cylinder, $V_\text{cylinder} = \pi r^2 h$, we substitute the values: $V_\text{cylinder} = 3.14 \times (13)^2 \times 26

Vcylinder=3.14×169×26V_\text{cylinder} = 3.14 \times 169 \times 26

Vcylinder=3.14×4394V_\text{cylinder} = 3.14 \times 4394

V_\text{cylinder} = 13790.36$ cubic inches So, the volume of the cylinder is approximately 13790.36 cubic inches. ### 5. Calculate the Empty Space The empty space inside the cylinder is the difference between the volume of the cylinder and the volume of the volleyball: $V_\text{empty} = V_\text{cylinder} - V_\text{sphere}

Vempty=13790.36−9212.83V_\text{empty} = 13790.36 - 9212.83

V_\text{empty} = 4577.53$ cubic inches ### 6. Round the Answer Rounding the empty space volume to the nearest whole number, we get: $V_\text{empty} \approx 4578$ cubic inches Therefore, the volume of the empty space inside the cylinder is approximately 4578 cubic inches. ## Final Answer After following the step-by-step solution, we have determined that the volume of the empty space inside the cylinder, not taken up by the volleyball, is approximately **_4578 cubic inches_**. This result was obtained by first calculating the volume of the volleyball (sphere) and the volume of the cylinder, and then subtracting the volume of the volleyball from the volume of the cylinder. Rounding the result to the nearest whole number gave us our final answer. This problem demonstrates a practical application of geometric formulas in calculating volumes and understanding spatial relationships. The process of breaking down the problem into smaller steps, applying the correct formulas, and performing the calculations systematically is a valuable skill in mathematics and other fields. ## Practice Problems To reinforce your understanding of the concepts discussed in this article, here are some practice problems you can try: 1. A basketball with a radius of 5 inches fits exactly inside a cylindrical container. Using $\pi = 3.14$, calculate the volume of the empty space inside the container. 2. A spherical ball has a radius of 8 cm. It is placed inside a cylinder with the same radius and height equal to the diameter of the sphere. Find the volume of the empty space in the cylinder. 3. A cylinder has a radius of 10 inches and a height of 20 inches. A sphere with a radius of 10 inches is placed inside the cylinder. Calculate the volume of the space not occupied by the sphere (use $\pi = 3.14$). These problems will help you practice applying the formulas for the volume of a sphere and a cylinder. Remember to break down each problem into steps, identify the given information, and use the appropriate formulas to find the solution. Practice makes perfect, so keep working on these problems to improve your skills. ## Conclusion In conclusion, we have successfully calculated the empty space within a cylinder that perfectly fits a volleyball. By applying the formulas for the volume of a sphere and a cylinder, we were able to determine that the volume of the empty space is approximately 4578 cubic inches. This problem highlights the practical application of geometry in everyday scenarios and reinforces the importance of understanding and applying mathematical formulas. The step-by-step approach we used—identifying the given information, determining the required formulas, performing the calculations, and rounding the answer—is a valuable method for solving a variety of mathematical problems. By breaking down complex problems into smaller, manageable steps, we can approach them with confidence and accuracy. Moreover, practicing similar problems, as suggested in the practice problems section, will further enhance your understanding and skills in geometry. Mathematics is a subject that builds upon itself, and a strong foundation in basic concepts is essential for tackling more advanced topics. Keep practicing, keep exploring, and you will continue to grow in your mathematical journey.