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Chapter 8: Problem 9

Find the volume of each sphere. $$r=0.90 \mathrm{m}$$

Short Answer

Expert verified
The volume of the sphere is approximately \( 3.06 \, \mathrm{m^3} \).

Step by step solution

01

Identify the formula for the volume of a sphere

The formula for the volume of a sphere is given by \[ V = \frac{4}{3} \, \pi \, r^3 \] where \( V \) is the volume and \( r \) is the radius of the sphere.
02

Substitute the radius into the formula

Given the radius \( r = 0.90 \, \mathrm{m} \), substitute \( r \) into the formula: \[ V = \frac{4}{3} \, \pi \, (0.90)^3 \]
03

Calculate the cube of the radius

Calculate \( (0.90)^3 \): \[ (0.90)^3 = 0.729 \]
04

Simplify the expression

Substitute \( 0.729 \) back into the volume formula: \[ V = \frac{4}{3} \, \pi \, (0.729) \]
05

Compute the volume

Calculate the volume: \[ V = \frac{4}{3} \, \pi \, (0.729) \] \[ V 鈮� 3.06 \, \mathrm{m^3} \]

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Key Concepts

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

sphere volume formula
To calculate the volume of a sphere, we use a specific formula that involves the radius and pi (蟺). The formula is written as: \[ V = \frac{4}{3} \, \pi \, r^3 \]This formula means that the volume (V) depends on the cube of the radius (r), multiplied by pi (蟺), and then multiplied by 4/3. It is important to follow this formula accurately to find the correct volume of any sphere.
radius
The radius of a sphere is the distance from the center of the sphere to any point on its surface. It is a key component in our volume formula. In the given problem, the radius is provided as 0.90 meters. When using the radius in calculations, always ensure it is in the correct units. For example, all measurements of length should be in meters if you are looking for a volume in cubic meters. Incorrect units will lead to inaccurate results.
pi
Pi, denoted as 蟺, is a mathematical constant that represents the ratio of a circle's circumference to its diameter. Pi is approximately equal to 3.14159. In our formula for the volume of a sphere, 蟺 plays a crucial role. It helps convert the geometric properties of the sphere into its volume. While you often see pi abbreviated as 3.14 or 22/7 in basic calculations, for more accurate results, especially in advanced problems, you should use more precise values of pi.
cube of a number
Cubing a number means raising it to the power of three, symbolized as \[ r^3 \]In the sphere's volume formula, the radius needs to be cubed to get the volume. For example, \((0.90^3 = 0.729)\). To calculate this, you multiply the number by itself three times:
  • 0.90 脳 0.90 = 0.81
  • 0.81 脳 0.90 = 0.729
This step is necessary to fully apply the volume formula accurately, which then gets multiplied by pi and 4/3 to give the sphere's volume.

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Most popular questions from this chapter

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