Question

The radius of a sphere is \(\frac{7}{2}\) cm. The volume of the sphere is:

• A. \(\frac{231}{3}\) cu cm
• B. \(\frac{539}{12}\) cu cm
• C. \(\frac{343}{6}\) cu cm
• D. \(154\) cu cm

Answer 1

The Correct Option is C

Step 1: Understanding the formula for the volume of a sphere:
The formula for the volume \(V\) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \] where \(r\) is the radius of the sphere.

Step 2: Substituting the given radius:
We are given that the radius of the sphere is \( r = \frac{7}{2} \) cm. Now, substitute this value into the volume formula:
\[ V = \frac{4}{3} \pi \left( \frac{7}{2} \right)^3 \] First, calculate \( \left( \frac{7}{2} \right)^3 \):
\[ \left( \frac{7}{2} \right)^3 = \frac{7^3}{2^3} = \frac{343}{8} \] Now, substitute this into the volume formula:
\[ V = \frac{4}{3} \pi \times \frac{343}{8} \] Simplifying the expression:
\[ V = \frac{4 \times 343}{3 \times 8} \pi = \frac{1372}{24} \pi \] Now, simplify \( \frac{1372}{24} \):
\[ \frac{1372}{24} = \frac{343}{6} \] Thus, the volume of the sphere is:
\[ V = \frac{343}{6} \pi \, \text{cm}^3 \]

Step 3: Conclusion:
The volume of the sphere is \( \frac{343}{6} \pi \) cubic centimeters.