Step 1: Surface Area of the Sphere
The formula for the surface area \( S \) of a sphere is:
\[ S = 4 \pi r^2 \] where \( r \) is the radius of the sphere. Given that the surface area \( S = 154 \, \text{cm}^2 \), we substitute this into the equation: \[ 154 = 4 \pi r^2 \]
Step 2: Solve for the Radius \( r \)
To solve for \( r \), divide both sides of the equation by \( 4 \pi \): \[ r^2 = \frac{154}{4 \pi} = \frac{154}{12.5664} \approx 12.25 \] Taking the square root of both sides: \[ r = \sqrt{12.25} \approx 3.5 \, \text{cm} \]
Step 3: Volume of the Sphere
The formula for the volume \( V \) of a sphere is: \[ V = \frac{4}{3} \pi r^3 \] Substituting \( r = 3.5 \, \text{cm} \) into this equation: \[ V = \frac{4}{3} \pi (3.5)^3 \] First, calculate \( (3.5)^3 \): \[ (3.5)^3 = 42.875 \] Now calculate the volume: \[ V = \frac{4}{3} \pi \times 42.875 \approx \frac{4}{3} \times 3.1416 \times 42.875 \approx 179.594 \, \text{cm}^3 \]
The volume of the sphere is approximately \( 179.59 \, \text{cm}^3 \).