A solid spherical wax mold with a radius of 21 cm is melted down and recast to form ‘n’ identical solid hemispherical domes. If the radius of each hemispherical dome is exactly half of the radius of the original spherical mold, find the sum of the total surface areas of all the hemispherical domes.

The correct answer is 16632 cm².

Step 1: Identify the given dimensions
Let the radius of the original solid spherical wax mold be $R=21\text{ cm}$.
The radius of each solid hemispherical dome is half of the original sphere's radius:
$r=\frac{R}{2}=\frac{21}{2}=10.5\text{ cm}$

Step 2: Find the number of hemispherical domes ($n$)
When a solid is melted down and recast into new shapes, the total volume of material remains unchanged. Therefore, the volume of the original sphere equals the total volume of the $n$ identical hemispherical domes.
Volume of the original sphere:
$V_{\text{sphere}}=\frac{4}{3}\pi R^3$
Volume of a single solid hemispherical dome:
$V_{\text{dome}}=\frac{2}{3}\pi r^3$
Equating the total volumes:
$n\times V_{\text{dome}}=V_{\text{sphere}}$
$n\times \frac{2}{3}\pi r^3=\frac{4}{3}\pi R^3$
Cancel common terms ($\frac{2}{3}\pi$) from both sides:
$n\times r^3=2R^3$
Substitute $r=\frac{R}{2}$:
$n\times {\left(\frac{R}{2}\right)}^3=2R^3$
$n\times \frac{R^3}{8}=2R^3$
Cancel $R^3$ from both sides:
$\frac{n}{8}=2$
$n=16$

Step 3: Calculate the total surface area of all hemispherical domes
The total surface area (TSA) of a single solid hemispherical dome includes its curved surface area ($2\pi r^2$) plus its flat circular base area ($\pi r^2$):
${\text{TSA}}_{\text{dome}}=3\pi r^2$
The total surface area of all $n=16$ domes is:
$\text{Total Area}=16\times 3\pi r^2=48\pi r^2$
Substitute $\pi =\frac{22}{7}$ and $r=\frac{21}{2}\text{ cm}$:
$\text{Total Area}=48\times \frac{22}{7}\times {\left(\frac{21}{2}\right)}^2$
$\text{Total Area}=48\times \frac{22}{7}\times \frac{441}{4}$
Simplifying the numbers:
$\text{Total Area}=\left(\frac{48}{4}\right)\times 22\times \left(\frac{441}{7}\right)$
$\text{Total Area}=12\times 22\times 63$
$\text{Total Area}=264\times 63=16632\text{ cm}{2}^{}$

Thus, the sum of the total surface areas of all the hemispherical domes is 16632 cm².