Answer: A. $209$

Explanation:1. Understand the geometry of a right square pyramid: Base is a square. Apex is directly above the center of the base. Surface Area $(SA) = \text{Area of Base} + \text{Lateral Surface Area}$. Lateral Surface Area $= 4 \times (\text{Area of one triangular face})$. Area of one triangular face $= \dfrac{1}{2} \times \text{base edge of triangle} \times \text{slant height}$. The base edge of the triangle is the side length of the square base.

2. Information given: Base area for both pyramids $= 100 \text{ cm}^2$. For a square base, if side length is $s$, then $s^2 = 100 \Rightarrow s = \sqrt{100} = 10$ cm. Pyramid 1: height $h\_1 = 11$ cm. Pyramid 2: height $h\_2 = 22$ cm.

3. Calculate the slant height ($l$) for each pyramid: The slant height is the hypotenuse of a right triangle formed by the pyramid's height ($h$) and half the base edge ($s/2$). $s/2 = 10/2 = 5$ cm. For Pyramid 1: $l\_1 = \sqrt{h\_1^2 + (s/2)^2} = \sqrt{11^2 + 5^2} = \sqrt{121 + 25} = \sqrt{146}$ cm. For Pyramid 2: $l\_2 = \sqrt{h\_2^2 + (s/2)^2} = \sqrt{22^2 + 5^2} = \sqrt{484 + 25} = \sqrt{509}$ cm.

4. Calculate the Lateral Surface Area (LSA) for each pyramid: LSA $= 4 \times (\dfrac{1}{2} \times s \times l) = 2 \times s \times l$. LSA$\_1 = 2 \times 10 \times \sqrt{146} = 20\sqrt{146}$ cm$^2$. LSA$\_2 = 2 \times 10 \times \sqrt{509} = 20\sqrt{509}$ cm$^2$.

5. Calculate the Total Surface Area (SA) for each pyramid: SA $= \text{Base Area} + \text{LSA}$. SA$\_1 = 100 + 20\sqrt{146}$ cm$^2$. SA$\_2 = 100 + 20\sqrt{509}$ cm$^2$.

6. Calculate the difference in surface areas: Difference $= SA\_2 - SA\_1 = (100 + 20\sqrt{509}) - (100 + 20\sqrt{146})$ Difference $= 100 + 20\sqrt{509} - 100 - 20\sqrt{146}$ Difference $= 20\sqrt{509} - 20\sqrt{146}$ Difference $= 20(\sqrt{509} - \sqrt{146})$

7. Approximate the values: $\sqrt{509} \approx 22.56$ (since $22^2 = 484, 23^2 = 529$) $\sqrt{146} \approx 12.08$ (since $12^2 = 144, 13^2 = 169$) Difference $\approx 20(22.56 - 12.08)$ Difference $\approx 20(10.48)$ Difference $\approx 209.6$

8. Compare with the given options: [A] $209$ [B] $220$ [C] $367$ [D] $551$ Our approximation $209.6$ is closest to $209$.

9. Conclusion: The difference in the surface area of the second pyramid and the surface area of the first pyramid is closest to $209 \text{ cm}^2$.