To find the surface area of a sphere given its volume, we can start with the formulas for the volume and surface area 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.
The surface area \( A \) of a sphere is given by: \[ A = 4 \pi r^2 \]
We are given that the volume \( V = 288\pi , \text{cm}^3 \). We can set the volume formula equal to this value and solve for the radius \( r \): \[ \frac{4}{3} \pi r^3 = 288\pi \]
Dividing both sides by \( \pi \): \[ \frac{4}{3} r^3 = 288 \]
Next, multiply both sides by \( \frac{3}{4} \): \[ r^3 = 288 \times \frac{3}{4} \] \[ r^3 = 216 \]
Now, taking the cube root of both sides to find \( r \): \[ r = \sqrt[3]{216} \]
Since \( 216 = 6^3 \), we find: \[ r = 6 , \text{cm} \]
Now that we have the radius, we can calculate the surface area \( A \): \[ A = 4 \pi r^2 \] Substituting \( r = 6 \): \[ A = 4 \pi (6^2) \] \[ A = 4 \pi (36) \] \[ A = 144 \pi , \text{cm}^2 \]
Therefore, the surface area of the sphere is: \[ \boxed{144\pi , \text{cm}^2} \]