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28. The volume of a sphere is 972π. What is the radius of the sphere? Work: radius=

28. The volume of a sphere is 972π. What is the radius of the sphere? Work: radius=

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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### Educational Content: Solving for the Radius of a Sphere --- #### Problem Statement: **28. The volume of a sphere is \(972\pi\). What is the radius of the sphere?** --- #### Explanation: To solve for the radius of a sphere when its volume is given: 1. **Volume Formula:** The volume \(V\) of a sphere can be found using the formula: \[ V = \frac{4}{3} \pi r^3 \] 2. **Given Data:** The volume of the sphere is \(972\pi\). 3. **Setting up the Equation:** Substitute the given volume into the volume formula: \[ 972\pi = \frac{4}{3} \pi r^3 \] 4. **Solving for \(r^3\):** Cancel out \(\pi\) from both sides: \[ 972 = \frac{4}{3} r^3 \] To isolate \(r^3\), multiply both sides by \(\frac{3}{4}\): \[ 972 \times \frac{3}{4} = r^3 \] Calculate: \[ 972 \times \frac{3}{4} = 729 \] So, \[ r^3 = 729 \] 5. **Finding the Radius \(r\):** Take the cube root of both sides to solve for \(r\): \[ r = \sqrt[3]{729} \] Calculate: \[ r = 9 \] 6. **Answer:** Hence, the radius of the sphere is \(9\) units. --- #### Final Answer: **radius = 9** This detailed solution explains the step-by-step approach to solving for the radius of a sphere when given its volume.
Transcribed Image Text:### Educational Content: Solving for the Radius of a Sphere --- #### Problem Statement: **28. The volume of a sphere is \(972\pi\). What is the radius of the sphere?** --- #### Explanation: To solve for the radius of a sphere when its volume is given: 1. **Volume Formula:** The volume \(V\) of a sphere can be found using the formula: \[ V = \frac{4}{3} \pi r^3 \] 2. **Given Data:** The volume of the sphere is \(972\pi\). 3. **Setting up the Equation:** Substitute the given volume into the volume formula: \[ 972\pi = \frac{4}{3} \pi r^3 \] 4. **Solving for \(r^3\):** Cancel out \(\pi\) from both sides: \[ 972 = \frac{4}{3} r^3 \] To isolate \(r^3\), multiply both sides by \(\frac{3}{4}\): \[ 972 \times \frac{3}{4} = r^3 \] Calculate: \[ 972 \times \frac{3}{4} = 729 \] So, \[ r^3 = 729 \] 5. **Finding the Radius \(r\):** Take the cube root of both sides to solve for \(r\): \[ r = \sqrt[3]{729} \] Calculate: \[ r = 9 \] 6. **Answer:** Hence, the radius of the sphere is \(9\) units. --- #### Final Answer: **radius = 9** This detailed solution explains the step-by-step approach to solving for the radius of a sphere when given its volume.
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