Answer :
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]
[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]
