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 option is 16632 cm².

Let's break down the solution step-by-step to find the sum of the total surface areas of all the hemispherical domes.

Step 1: Understand the Given Data
Radius of the solid spherical wax mold, R = 21 cm.
Radius of each solid hemispherical dome, r = half of the radius of the original sphere = 21 / 2 cm = 10.5 cm.

Step 2: Find the Number of Hemispherical Domes (n)
When the spherical mold is melted and recast into 'n' identical hemispherical domes, the total volume remains conserved.

Volume of the original solid sphere:

$V_{\text{sphere}}=\frac{4}{3}\pi R^3$

Volume of one solid hemisphere:

$V_{\text{hemisphere}}=\frac{2}{3}\pi r^3$

Equating the volumes:

$V_{\text{sphere}}=n\times V_{\text{hemisphere}}$

$\frac{4}{3}\pi R^3=n\times \frac{2}{3}\pi r^3$

Simplifying the equation by canceling $\frac{2}{3}\pi$ from both sides:

$2R^3=n\times r^3$

Since $r=\frac{R}{2}$, substitute r into the equation:

$2R^3=n\times {\left(\frac{R}{2}\right)}^3$

$2R^3=n\times \frac{R^3}{8}$

Canceling R³ from both sides gives:

$n=2\times 8=16$

So, 16 hemispherical domes are formed.

Step 3: Calculate the Total Surface Area of One Hemispherical Dome
The total surface area (TSA) of a solid hemisphere includes its curved surface area plus the area of its flat circular base:

$\text{TSA of one hemisphere}=3\pi r^2$

Step 4: Calculate the Sum of Total Surface Areas of All 'n' Domes

$\text{Sum of total surface areas}=n\times 3\pi r^2$

Substitute n = 16, π = 22 / 7, and r = 21 / 2 cm:

$\text{Sum of total surface areas}=16\times 3\times \frac{22}{7}\times {\left(\frac{21}{2}\right)}^2$

$\text{Sum of total surface areas}=16\times 3\times \frac{22}{7}\times \frac{441}{4}$

Simplify by canceling terms (16 divided by 4 gives 4, and 441 divided by 7 gives 63):

$\text{Sum of total surface areas}=4\times 3\times 22\times 63$

$\text{Sum of total surface areas}=12\times 1386=16632{\text{cm}}^2$

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