Convert Mixed Fraction to Improper Fraction

Convert the given surface area from a mixed fraction into an improper fraction: $2124\frac{4}{7} = \frac{2124 \times 7 + 4}{7} = \frac{14868 + 4}{7} = \frac{14875}{7}\text{ cm}^2$.

$S = \frac{14875}{7} \text{ cm}^2$

Equate Surface Area to Formula

The surface area of a sphere is given by $S = 4\pi r^2$. Substitute $S = \frac{14875}{7}$ and $\pi = \frac{22}{7}$.

$4 \times \frac{22}{7} \times r^2 = \frac{14875}{7}$

Solve for Radius ($r$)

Simplify the equation to find $r^2$: $\frac{88}{7} r^2 = \frac{14875}{7} \implies 88 r^2 = 14875 \implies r^2 = \frac{14875}{88} = \frac{1352.27...}{...} \text{ wait, let's divide properly: } 14875 / 88 = 169...$ Let's re-verify: $169 \times 88 = 14872$? Let's check $14875 / 88$: wait, $2124 \frac{4}{7} = \frac{14872 + 4}{7} = \frac{14876}{7}$. Ah, $2124 \times 7 = 14868$, plus $4 = 14872$. Wait, the question says $2124\frac{4}{7}$, let's check $2124 \times 7 = 14868$, plus $4 = 14872 / 7$. Then $4 \times \frac{22}{7} \times r^2 = \frac{14872}{7} \implies \frac{88}{7} r^2 = \frac{14872}{7} \implies 88 r^2 = 14872 \implies r^2 = \frac{14872}{88} = 169$. Thus $r = \sqrt{169} = 13\text{ cm}$.

$r^2 = 169 \implies r = 13 \text{ cm}$

Calculate Diameter ($d$)

The diameter of the sphere is twice the radius: $d = 2r = 2 \times 13\text{ cm} = 28\text{ cm}$. Wait, $2 \times 13 = 26\text{ cm}$?! Let's re-multiply: radius $r = 13$, diameter $= 2 \times 13 = 26\text{ cm}$. Wait, let's check option D: $14$\text$