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Chapter 10: Problem 5

Find the volume of a sphere whose surface area is \(64 \pi \mathrm{cm}^{2}\) (IMAGE CAN'T COPY)

Short Answer

Expert verified
\[ \frac{256}{3} \pi \mathrm{cm}^3 \]

Step by step solution

01

Recall the Formula for Surface Area

The surface area (SA) of a sphere is given by the formula: \[ SA = 4 \pi r^2 \]
02

Solve for the Radius

Given that the surface area is 64 \( \pi \) \mathrm{cm}^{2}, set up the equation: \[ 4 \pi r^2 = 64 \pi \]Next, divide both sides by \( 4 \pi \): \[ r^2 = 16 \]Take the square root of both sides to solve for \( r \): \[ r = 4 \text{ cm} \]
03

Recall the Formula for Volume

The volume (V) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \]
04

Substitute the Radius into the Volume Formula

Substitute \( r = 4 \) cm into the volume formula: \[ V = \frac{4}{3} \pi (4)^3 \]Calculate the value inside the parentheses: \[ V = \frac{4}{3} \pi (64) \]Finally, multiply: \[ V = \frac{4}{3} \pi \times 64 = \frac{256}{3} \pi \]
05

State the Volume

Thus, the volume of the sphere is: \[ V = \frac{256}{3} \pi \mathrm{cm}^3 \]

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Key Concepts

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

Volume of a Sphere
A sphere is a perfectly round 3-dimensional shape where every point on the surface is equidistant from the center. The volume of a sphere is the total space inside it. To find this, use the formula \( V = \frac{4}{3} \pi r^3 \). In this formula, \( V \) represents the volume, \( \pi \) (pi) is approximately 3.14159, and \( r \) is the radius of the sphere. Let's break it down:
  • First, you need the radius, which is the distance from the center to the surface.
  • Then, cube the radius, which means multiplying it by itself twice (\( r \tiny^3 = r \times r \times r \)).
  • Next, multiply by \( \pi \).
  • Finally, multiply by \( \frac{4}{3} \).
This formula helps in calculating how much space a sphere occupies. In our example, with a radius of 4 cm, the volume calculation goes like this: \( V = \frac{4}{3} \pi (4)^3 = \frac{256}{3} \pi \mathrm{cm}^3 \). This means the volume of the sphere is \frac{256}{3} \pi \mathrm{cm}^3.
Surface Area
Surface area is the total area that the surface of an object occupies. For a sphere, this is calculated by the formula \( SA = 4 \pi r^2 \). Here, \( SA \) stands for surface area, and \( r \) is the radius. Let鈥檚 see how it works:
  • First, square the radius (\( r^2 = r \times r \)).
  • Then, multiply the result by 4.
  • Lastly, multiply by \( \pi \).
In our specific case, the surface area is given as \( 64 \pi \mathrm{cm}^2 \). We can use this information to find the radius: \( 4 \times \pi \times r^2 = \64 \pi \). By simplifying, we find \( r=4 \mathrm{cm} \). This radius can now be used to find the volume, as shown earlier.
Formulas
Formulas are essential in geometry and other areas of mathematics. They provide a shortcut to solve problems efficiently. Here are the two key formulas used in this exercise:
  • **Surface Area of a Sphere:** \( SA = 4 \pi r^2 \)
  • **Volume of a Sphere:** \( V = \frac{4}{3} \pi r^3 \)
These formulas relate the radius with the surface area and volume. By knowing one, you can often find the other, making them powerful tools in geometry problem-solving. Recalling and understanding these formulas will help you tackle a variety of geometry problems involving spheres. It's like having a mathematical toolkit.

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Most popular questions from this chapter

Identify each statement as true or false. Sketch a counterexample for each false statement or explain why it is false. Every section of a cylinder, parallel to the base, is congruent to the base.

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Application An auto tunnel through a mountain is being planned. It will be in the shape of a semicircular cylinder with a radius of \(10 \mathrm{m}\) and a length of \(2 \mathrm{km}\). How many cubic meters of dirt will need to be removed? If the bed of each dump truck has dimensions \(2.2 \mathrm{m}\) by \(4.8 \mathrm{mby} 1.8 \mathrm{m},\) how many loads will be required to carry away the dirt?

Identify each statement as true or false. Sketch a counterexample for each false statement or explain why it is false. A lateral face of a pyramid is always a triangular region.

The base of a hemisphere has an area of \(256 \pi \mathrm{cm}^{2}\) Find its volume.
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