Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To find the volume of the sphere, let’s start by understanding the relationship between the volumes of the cylinder and the sphere given their geometrical properties.
1. Volume of a Cylinder Formula:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
Given:
[tex]\[ V_{\text{cylinder}} = 21 \text{ m}^3 \][/tex]
2. Volume of a Sphere Formula:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \][/tex]
Since the height of the cylinder [tex]\( h \)[/tex] equals the diameter of the sphere, which is [tex]\( 2r \)[/tex], we can link the cylinder's volume to the sphere's radius:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \][/tex]
Given:
[tex]\[ 21 = 2\pi r^3 \][/tex]
Now, we solve for [tex]\( r^3 \)[/tex]:
[tex]\[ r^3 = \frac{21}{2\pi} \][/tex]
Next, we use the volume formula for the sphere:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \][/tex]
Substitute [tex]\( r^3 \)[/tex] from the previous step:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{21}{2\pi} \right) \][/tex]
Simplify:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \cdot \frac{21}{2} \][/tex]
[tex]\[ V_{\text{sphere}} = \frac{4 \cdot 21}{3 \cdot 2} = \frac{84}{6} = 14 \][/tex]
Therefore, the volume of the sphere is:
[tex]\[ 14 \text{ m}^3 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{14 \text{ m}^3} \][/tex]
1. Volume of a Cylinder Formula:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
Given:
[tex]\[ V_{\text{cylinder}} = 21 \text{ m}^3 \][/tex]
2. Volume of a Sphere Formula:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \][/tex]
Since the height of the cylinder [tex]\( h \)[/tex] equals the diameter of the sphere, which is [tex]\( 2r \)[/tex], we can link the cylinder's volume to the sphere's radius:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h = \pi r^2 (2r) = 2\pi r^3 \][/tex]
Given:
[tex]\[ 21 = 2\pi r^3 \][/tex]
Now, we solve for [tex]\( r^3 \)[/tex]:
[tex]\[ r^3 = \frac{21}{2\pi} \][/tex]
Next, we use the volume formula for the sphere:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \][/tex]
Substitute [tex]\( r^3 \)[/tex] from the previous step:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi \left( \frac{21}{2\pi} \right) \][/tex]
Simplify:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \cdot \frac{21}{2} \][/tex]
[tex]\[ V_{\text{sphere}} = \frac{4 \cdot 21}{3 \cdot 2} = \frac{84}{6} = 14 \][/tex]
Therefore, the volume of the sphere is:
[tex]\[ 14 \text{ m}^3 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{14 \text{ m}^3} \][/tex]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.