Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.

The radius of a sphere is 6 units.

Which expression represents the volume of the sphere, in cubic units?

A. [tex] \frac{3}{4} \pi(6)^2 [/tex]
B. [tex] \frac{4}{3} \pi(6)^3 [/tex]
C. [tex] \frac{3}{4} \pi(12)^2 [/tex]
D. [tex] \frac{4}{3} \pi(12)^3 [/tex]


Sagot :

To determine which expression represents the volume of a sphere with a radius of 6 units, we need to use the formula for the volume of a sphere:

[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]

Given \( r = 6 \), substituting this into the formula gives:

[tex]\[ V = \frac{4}{3} \pi (6)^3 \][/tex]

Let's assess each of the given expressions one by one to see if they match this formula.

1. Expression: \( \frac{3}{4} \pi (6)^2 \)
- This expression calculates the area of a circle (not the volume of a sphere), but adjusted by the fraction.
- Volume check: \(\frac{3}{4} \pi (6)^2 = \frac{3}{4} \pi (36) = 27 \pi\)

2. Expression: \( \frac{4}{3} \pi (6)^3 \)
- This is the exact formula for the volume of a sphere with radius 6.
- Volume calculation: \( \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi\)

3. Expression: \( \frac{3}{4} \pi (12)^2 \)
- This calculates the area of a circle with diameter 12, adjusted by the fraction.
- Volume check: \(\frac{3}{4} \pi (12)^2 = \frac{3}{4} \pi (144) = 108 \pi\)

4. Expression: \( \frac{4}{3} \pi (12)^3 \)
- This calculates the volume of a sphere with radius 12, not 6.
- Volume check: \(\frac{4}{3} \pi (12)^3 = \frac{4}{3} \pi (1728) = 2304 \pi\)

From our evaluations, only the expression \(\frac{4}{3} \pi (6)^3\) matches both the necessary formula and the correct radius value.

Thus, the correct expression representing the volume of the sphere is:

[tex]\[ \frac{4}{3} \pi (6)^3 \][/tex]