To find the volume of a sphere with a given radius, we use the formula for the volume of a sphere:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Here, [tex]\( r \)[/tex] represents the radius of the sphere. In this case, the radius [tex]\( r \)[/tex] is given as 5.
Substituting the value of the radius into the formula, we get:
[tex]\[ V = \frac{4}{3} \pi (5^3) \][/tex]
We can verify which of the given expressions matches this formula.
A. [tex]\(\frac{4}{3} \pi\left(5^2\right)\)[/tex]
B. [tex]\(4 \pi\left(5^2\right)\)[/tex]
C. [tex]\(4 \pi\left(5^3\right)\)[/tex]
D. [tex]\(\frac{4}{3} \pi\left(5^3\right)\)[/tex]
Upon inspection:
- Option A uses [tex]\( 5^2 \)[/tex] rather than [tex]\( 5^3 \)[/tex], so it is incorrect.
- Option B uses [tex]\( 5^2 \)[/tex] rather than [tex]\( 5^3 \)[/tex] and it is multiplied by 4 instead of [tex]\(\frac{4}{3}\)[/tex], so it is incorrect.
- Option C is [tex]\( 4 \pi (5^3) \)[/tex] which misses the [tex]\(\frac{4}{3}\)[/tex] factor, so it is incorrect.
- Option D uses the correct exponent [tex]\( 5^3 \)[/tex] and the correct factor of [tex]\(\frac{4}{3}\)[/tex], making it the correct expression.
Thus, the correct expression that gives the volume of a sphere with radius 5 is:
[tex]\[ \boxed{\frac{4}{3} \pi (5^3)} \][/tex]
