Question

the volume of the sphere is \\(\frac{500}{3}\pi\\) cubic units. what is the value of \\(x\\)? \\(\bigcirc\\) 4 units \\(\bigcirc\\) 5 units \\(\bigcirc\\) 8 units \\(\bigcirc\\) 10 units

Explanation:

Step1: Recall the volume formula for a sphere

The volume \( V \) of a sphere is given by the formula \( V=\frac{4}{3}\pi r^{3} \), where \( r \) is the radius of the sphere. Here, the radius of the sphere is \( x \), so we have \( V = \frac{4}{3}\pi x^{3} \).

Step2: Substitute the given volume into the formula

We know that the volume \( V=\frac{500}{3}\pi \). Substituting this into the volume formula, we get:
\[
\frac{4}{3}\pi x^{3}=\frac{500}{3}\pi
\]

Step3: Solve for \( x^{3} \)

First, we can divide both sides of the equation by \( \pi \) (since \( \pi
eq0 \)) to simplify:
\[
\frac{4}{3}x^{3}=\frac{500}{3}
\]
Then, multiply both sides by \( \frac{3}{4} \) to isolate \( x^{3} \):
\[
x^{3}=\frac{500}{3}\times\frac{3}{4}
\]
The \( 3 \) in the numerator and denominator cancels out, so we have:
\[
x^{3}=\frac{500}{4}=125
\]

Step4: Solve for \( x \)

We need to find the cube root of \( 125 \) to get \( x \). Since \( 5\times5\times5 = 125 \), we have \( x=\sqrt[3]{125}=5 \).

Answer:

5 units (corresponding to the option "5 units")