Answered

Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.

A sphere and a cylinder have the same radius and height. The volume of the cylinder is [tex]21 \, \text{m}^3[/tex].

What is the volume of the sphere?

A. [tex]6 \, \text{m}^3[/tex]
B. [tex]7 \, \text{m}^3[/tex]
C. [tex]14 \, \text{m}^3[/tex]

Sagot :

GinnyAnswer
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]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.