Sphere Volume Numerically Equals Surface Area

This problem asks for the radius and surface area of a sphere where its volume is numerically equal to its surface area, using \(\pi = \frac{22}{7}\) and rounding to 3 decimal places.

Sphere Formulas and Given Values

• Volume (\(V\)) of a sphere: \(\small V = \frac{4}{3}\pi r^3\)
• Surface Area (\(A\)) of a sphere: \(\small A = 4\pi r^2\)
• Given value of pi: \(\pi = \frac{22}{7}\)
• Condition: Volume = Surface Area (numerically)

Equating Sphere Volume and Surface Area

The condition \(V = A\) translates to:

\( \frac{4}{3}\pi r^3 = 4\pi r^2 \)

Sphere Radius Calculation

To solve for the radius (\(r\)), we simplify the equation. Assuming \(r > 0\) (since it's a sphere):

Divide both sides by \(4\pi r^2\):

\( \frac{\frac{4}{3}\pi r^3}{4\pi r^2} = \frac{4\pi r^2}{4\pi r^2} \)

This simplifies to:

\( \frac{1}{3}r = 1 \)

Solving for \(r\) yields:

\( r = 3 \text{ units} \)

Sphere Surface Area Calculation

Using the calculated radius (\(r = 3\)) and the given \(\pi = \frac{22}{7}\), calculate the surface area (\(A\)):

\( A = 4\pi r^2 \)

\( A = 4 \times \frac{22}{7} \times (3)^2 \)

\( A = 4 \times \frac{22}{7} \times 9 \)

\( A = \frac{4 \times 22 \times 9}{7} \)

\( A = \frac{792}{7} \)

Converting to a decimal and rounding to three decimal places:

\( A \approx 113.143 \text{ unit}^2 \)

Final Results

The radius of the sphere is 3 units, and its surface area is approximately 113.143 unit\(^2\).