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Chapter 12: Problem 1

Find the volume of a sphere with a. A radius of 3 b. A diameter of 18 c. A radius of 5

Short Answer

Expert verified
The volumes are \(36\pi\) cubic units for a sphere with a radius of 3, \(972\pi\) cubic units for a sphere with a diameter of 18, and \(\frac{500\pi}{3}\) cubic units for a sphere with a radius of 5.

Step by step solution

01

Volume of Sphere with Radius 3

Use the formula for the volume of a sphere to find its size. The radius is 3, so substitute this value into the formula. The calculation is as follows: \(Volume = \frac{4}{3}\pi(3)^3 = 36\pi\) cubic units.
02

Volume of Sphere with Diameter 18

Let's find the volume of a sphere with a diameter of 18. However, the formula needs the radius of the sphere, so begin by dividing the diameter by 2 to obtain the radius (18/2 = 9). Apply the volume formula with r=9: \(Volume = \frac{4}{3}\pi(9)^3 = 972\pi\) cubic units.
03

Volume of Sphere with Radius 5

Finally, calculate the volume of a sphere with a radius of 5. Substituting in the radius, we get \(Volume = \frac{4}{3}\pi(5)^3 = \frac{500\pi}{3}\) cubic units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometry Formulas
Understanding the tools that are used to tackle geometry problems is crucial. The formula for calculating the volume of a sphere is an important one. It assures that you get accurate results when determining how much space a spherical object occupies. The formula is expressed as:
  • \( Volume = \frac{4}{3}\pi r^3 \)
The symbol \( \pi \) (Pi) represents approximately 3.14159. The letter \( r \) stands for the radius of the sphere, which is the distance from the center of the sphere to any point on its surface. Employing this formula precisely allows us to compute the volume of a sphere when the radius is known.
Mathematics Education
In the realm of mathematics education, learning how to apply formulas like the one for sphere volumes is foundational. Teaching these concepts helps students not only solve geometry problems but also strengthens their problem-solving skills overall. Encouraging students to break down complex problems into manageable steps fosters a deeper understanding.
  • Begin by identifying the problem parameters, such as knowing whether you have a radius or diameter.
  • Connect the dots by recalling relevant formulas and practicing their application.
This enhances the student's ability to approach similar problems logically and effectively.
Geometry Problems
Geometry problems often require a multi-step approach for solutions. This is evident in our example where the aim is to find the volume of spheres with different dimensions. Let鈥檚 summarize how to tackle these problems:
  • Identify key values, such as the radius or diameter.
  • If given the diameter, remember to convert it to a radius by dividing by 2.
  • Substitute the value of the radius into the volume formula \( \frac{4}{3}\pi r^3 \).
By following these steps, we maintain clarity and accuracy in our solutions, ensuring better comprehension of geometry concepts.
Volume Calculation
Calculating volume, especially of a sphere, involves understanding a few key principles. Start by ensuring that you have the radius, as it is essential for using the formula accurately. In practice problems, like those solved in the exercise, we see how:
  • For a radius of 3, the volume calculates to \( 36\pi\) cubic units.
  • For a diameter of 18, first convert to a radius of 9, leading to \( 972\pi \) cubic units.
  • For a radius of 5, the resulting volume is \( \frac{500\pi}{3} \) cubic units.
These examples demonstrate how understanding and applying the sphere volume formula accurately leads to successful calculation and offers a practical guide for finding sphere volumes in any setting.

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Most popular questions from this chapter

a Find the volume of a cube with an edge of 7 . b Find the volume of a cube with an edge of e. c Find the edge of a cube with a volume of \(125 .\) (Figure can't copy)

Traci's queen-size waterbed is 7 ft long. 5 ft wide, and 8 in. thick. a Find the bed's volume to the nearest cubic foot. b If 1 cu ft of water weighs 62.4 Ib, what is the weight of the water in Traci's bed to the nearest pound? (Figure can't copy)

Find the ratio of the volume of a sphere to the volume of the smallest right cylinder that can contain it.

A rubber ball is formed by a rubber shell filled with air. The shell's outer diameter is \(48 \mathrm{mm},\) and its inner diameter is \(42 \mathrm{mm} .\) Find, to the nearest cubic centimeter, the volume of rubber used to make the ball.

The perimeter of the scalene base of a pentagonal right prism is 17, and a lateral edge of the prism measures 10. Find the prism's lateral area.
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