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A sphere has a diameter of [tex]$14 \, \text{ft}$[/tex]. Which equation finds the volume of the sphere?

A. [tex] V = \frac{4}{3} (7)^3 [/tex]
B. [tex] V = \frac{4}{3} \pi (7)^3 [/tex]
C. [tex] V = \frac{4}{3} (14)^3 [/tex]
D. [tex] V = \frac{4}{3} \pi (14)^3 [/tex]

Sagot :

To find the volume of a sphere given its diameter, follow these steps:

1. Determine the Radius: The diameter of the sphere is given as [tex]\( 14 \)[/tex] feet. The radius of a sphere is half its diameter. Hence,
[tex]\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{14}{2} = 7 \, \text{feet} \][/tex]

2. Write the Formula for the Volume of a Sphere: The formula to find the volume [tex]\( V \)[/tex] of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
where [tex]\( r \)[/tex] is the radius of the sphere.

3. Substitute the Radius into the Volume Formula: Now, substitute the value of the radius [tex]\( r = 7 \)[/tex] feet into the formula:
[tex]\[ V = \frac{4}{3} \pi (7)^3 \][/tex]

4. Calculate [tex]\( 7^3 \)[/tex]: First, find the cube of the radius:
[tex]\[ 7^3 = 7 \times 7 \times 7 = 343 \][/tex]

5. Plug the Cubed Radius Back into the Volume Formula:
[tex]\[ V = \frac{4}{3} \pi (343) \][/tex]

6. Simplified Equation: This gives us the equation for the volume of the sphere:
[tex]\[ V = \frac{4}{3} \pi (7)^3 \][/tex]

Therefore, the correct equation that finds the volume of the sphere is:
[tex]\[ \boxed{V = \frac{4}{3} \pi (7)^3} \][/tex]
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