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Magazine Chapter 9: Problem 14
Find the volume of a sphere whose radius is 9 inches. Use \(\pi=3.14\). (a) 28.26 in. \(^{3}\) (b) 3052.08 in. \(^{3}\) (c) 972 in. \(^{3}\) (d) 2289.06 in. \(^{3}\)
Short Answer
Expert verified
The volume of the sphere is 3052.08 in. \(^3\), which corresponds to option (b).
Step by step solution
01
Formula for Volume of a Sphere
To find the volume of a sphere, we use the formula \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the sphere and \( \pi \) is approximately 3.14.
02
Substituting the Radius
Substitute the given radius of 9 inches into the formula. Replace \( r \) with 9 in the formula: \( V = \frac{4}{3} \times 3.14 \times 9^3 \).
03
Calculating \( r^3 \)
First calculate \( 9^3 \). This equals \( 9 \times 9 \times 9 = 729 \).
04
Applying the Formula
Insert \( 729 \) back into the formula for volume. The expression becomes \( V = \frac{4}{3} \times 3.14 \times 729 \).
05
Simplifying the Multiplication
Calculate the multiplication \( 3.14 \times 729 \). This equals \( 2289.06 \).
06
Multiplying by \( \frac{4}{3} \)
Finally, multiply \( 2289.06 \) by \( \frac{4}{3} \). This gives us the volume \( 3052.08 \).
07
Selecting the Correct Answer
Compare the calculated volume with the options provided. The correct answer is (b) 3052.08 in. \(^3\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Volume of a Sphere
The volume of a sphere is a measure of the amount of space it occupies. Imagine a perfectly round ball; figuring out how much air or material can fill up that ball is what we refer to as its volume. To find the volume of this sphere, a special formula is used:
- The Formula: The formula to calculate a sphere's volume is \( V = \frac{4}{3} \pi r^3 \).
- What it Means: Here, \( V \) is the volume, \( \pi \) is a mathematical constant (around 3.14), and \( r \) is the radius of the sphere.
Mathematics Formula
In mathematics, formulas are like recipes. They guide us to solve specific problems efficiently and accurately. The formula \( V = \frac{4}{3} \pi r^3 \) for finding the volume of a sphere is a great example of this.
- Breaking it Down: \( \frac{4}{3} \) is a fractional constant that ensures the formula gives the right measure of volume for a spherical shape.
- Understanding \( \pi \): \( \pi \), which is approximately 3.14, is a universal constant used in various calculations involving circles and spheres. It represents the ratio of a circle's circumference to its diameter.
- Exponentiation: The \( r^3 \) portion of the formula stands for \( r \times r \times r \), multiplying the radius by itself two more times.
Pre-Algebra
Pre-algebra is an important foundational stage in mathematics. It prepares students for understanding algebraic concepts by focusing on basic arithmetic and the introduction of variables.
- Working with Numbers: Pre-algebra often covers operations such as addition, subtraction, multiplication, and division, as seen in our calculation for the volume of a sphere.
- Understanding Variables: Here, \( r \) is a variable representing the radius. Pre-algebra helps students understand how to use such symbols to stand in for numbers.
- Applying Arithmetic: In finding the sphere's volume, we perform arithmetic operations like exponentiation (\( r^3 \)), and multiplication, similar to solving basic pre-algebra problems.
Radius of a Sphere
The radius of a sphere is crucial for calculating its volume. The radius (
- What is the Radius? The radius is the distance from the center of the sphere to any point on its surface. It's half of the diameter.
- Its Role in Volume Calculation: In our volume formula \( V = \frac{4}{3} \pi r^3 \), the radius is raised to the third power, highlighting its significant impact on the size or space the sphere can occupy.
- Units Matter: Ensure the radius is in the same units as required for the volume formula (e.g., inches, centimeters) to maintain consistency in calculations.
