Answer: A. $48$

Explanation:1. Understand the components of the total surface area of a pyramid: Total Surface Area (TSA) = Area of Base + Lateral Surface Area (LSA). Given: TSA = 36,864 sq inches. LSA (combined surface area of the four lateral faces) = 20,480 sq inches.

2. Calculate the area of the base: Area of Base = TSA - LSA Area of Base = 36,864 - 20,480 Area of Base = 16,384 sq inches.

3. Determine the side length of the square base: Let the side length of the square base be 'b'. Area of Base = $b^2$. $b^2 = 16,384$ $b = \sqrt{16384}$ To find $\sqrt{16384}$: Notice $100^2 = 10000$, $130^2 = 16900$. So b is likely between 100 and 130. The number ends in 4, so its square root must end in 2 or 8. Let's try numbers ending in 8 near 130, e.g., 128. $128^2 = (128)(128)$ $128 \times 100 = 12800$ $128 \times 20 = 2560$ $128 \times 8 = 1024$ Sum = $12800 + 2560 + 1024 = 15360 + 1024 = 16384$. So, $b = 128$ inches.

4. Relate LSA to slant height and base perimeter: LSA of a pyramid = $\dfrac{1}{2} \times \text{Perimeter of Base} \times \text{Slant Height (l)}$. Perimeter of Base = $4b = 4 \times 128 = 512$ inches. LSA = 20,480 sq inches. $20480 = \dfrac{1}{2} \times 512 \times l$ $20480 = 256 \times l$ $l = \dfrac{20480}{256}$ To calculate $20480 / 256$: $256 \times 100 = 25600$ (too high) Try $256 \times 80 = 256 \times 8 \times 10$. $256 \times 8 = (250 \times 8) + (6 \times 8) = 2000 + 48 = 2048$. So, $256 \times 80 = 20480$. Thus, slant height $l = 80$ inches.

5. Use the relationship between height (h), slant height (l), and half the base side length: In a right square pyramid, the height (h), the slant height (l), and the apothem of the base (which is half the side length of the square base, $\dfrac{b}{2}$) form a right-angled triangle. The slant height is the hypotenuse. $h^2 + \left(\dfrac{b}{2}\right)^2 = l^2$ We have $b=128$, so $\dfrac{b}{2} = \dfrac{128}{2} = 64$ inches. We have $l=80$ inches. $h^2 + (64)^2 = (80)^2$ $h^2 + 4096 = 6400$ $h^2 = 6400 - 4096$ $h^2 = 2304$

6. Calculate the height h: $h = \sqrt{2304}$ We need a number whose square is 2304. Consider numbers ending in 2 or 8. $40^2 = 1600$, $50^2 = 2500$. So h is between 40 and 50. Try 48: $48^2 = (50-2)^2 = 50^2 - 2(50)(2) + 2^2 = 2500 - 200 + 4 = 2304$. So, $h = 48$ inches.

7. Check the options: Option [A] is 48, which matches our result.

8. Conclusion: The height of the pyramid is $48$ inches. Option [A] is correct.