Question

if a sphere has volume \\(\frac{4}{3}\pi\\) cubic yards, what is its radius?

Explanation:

Step1: Recall the volume formula of 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.

Step2: Substitute the given volume into the formula

We are given that \( V = \frac{4}{3}\pi \). Substituting this into the volume formula, we get:

$$ \frac{4}{3}\pi=\frac{4}{3}\pi r^{3} $$

Step3: Solve for \( r \)

First, we can divide both sides of the equation by \( \frac{4}{3}\pi \).
Dividing both sides by \( \frac{4}{3}\pi \), we have:

$$ \frac{\frac{4}{3}\pi}{\frac{4}{3}\pi}=r^{3} $$

Simplifying the left - hand side, \( \frac{\frac{4}{3}\pi}{\frac{4}{3}\pi} = 1 \), so \( r^{3}=1 \).
Taking the cube root of both sides, since \( \sqrt[3]{1}=1 \), we get \( r = 1 \).

Answer:

The radius of the sphere is \( 1 \) yard.