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To solve the problem, we need to determine the height of the pyramid based on the given information.

### Information provided:
- The total surface area of the pyramid is **36,864 square inches**.
- The combined surface area of the four lateral faces is **20,480 square inches**.
- The pyramid is a **right square pyramid**, which means its base is a square and the lateral faces are triangles.

### Step-by-step solution:

1. **Surface Area Formula for a Right Square Pyramid**:
   The surface area \( A_{\text{total}} \) of a right square pyramid is the sum of the area of its square base and the areas of its four triangular lateral faces:
   \[
   A_{\text{total}} = A_{\text{base}} + A_{\text{lateral}}
   \]
   where:
   - \( A_{\text{base}} \) is the area of the square base.
   - \( A_{\text{lateral}} \) is the combined area of the four triangular faces.

   According to the problem, \( A_{\text{total}} = 36,864 \, \text{in}^2 \) and \( A_{\text{lateral}} = 20,480 \, \text{in}^2 \).

2. **Find the Area of the Base**:
   Using the surface area formula:
   \[
   A_{\text{base}} = A_{\text{total}} - A_{\text{lateral}}
   \]
   \[
   A_{\text{base}} = 36,864 - 20,480 = 16,384 \, \text{in}^2
   \]
   Since the base is a square, if \( s \) is the side length of the square:
   \[
   A_{\text{base}} = s^2
   \]
   So:
   \[
   s^2 = 16,384
   \]
   Taking the square root:
   \[
   s = \sqrt{16,384} = 128 \, \text{in}
   \]
   Therefore, the side length of the base is **128 inches**.

3. **Find the Height of the Pyramid**:
   The lateral surface area consists of four triangular faces, and the area of one triangular face is:
   \[
   A_{\text{triangle}} = \frac{1}{2} \times \text{base of the triangle} \times \text{slant height}
   \]
   where the base of the triangle is the side of the square \( s = 128 \, \text{in} \).

   The combined area of the four triangles is:
   \[
   A_{\text{lateral}} = 4 \times \left( \frac{1}{2} \times 128 \times \text{slant height} \right)
   \]
   Simplifying this:
   \[
   A_{\text{lateral}} = 2 \times 128 \times \text{slant height} = 256 \times \text{slant height}
   \]
   But we know \( A_{\text{lateral}} = 20,480 \, \text{in}^2 \), so:
   \[
   256 \times \text{slant height} = 20,480
   \]
   Solving for the slant height:
   \[
   \text{slant height} = \frac{20,480}{256} = 80 \, \text{in}
   \]

4. **Relate Slant Height to the Height of the Pyramid**:
   In a right square pyramid, the slant height \( l \), height \( h \), and half of the base \( \frac{s}{2} \) form a right triangle. Using the Pythagorean theorem:
   \[
   l^2 = h^2 + \left( \frac{s}{2} \right)^2
   \]
   Substituting the known values:
   \[
   80^2 = h^2 + \left( \frac{128}{2} \right)^2
   \]
   \[
   6,400 = h^2 + 64^2
   \]
   \[
   6,400 = h^2 + 4,096
   \]
   Solving for \( h^2 \):
   \[
   h^2 = 6,400 - 4,096 = 2,304
   \]
   Taking the square root:
   \[
   h = \sqrt{2,304} = 48 \, \text{in}
   \]

Thus, the height of the pyramid is **48 inches**.

### Answer:
The correct option is **A) 48**.

Would you like further clarification or details on any part of the solution?

### Related questions:
1. How do you calculate the volume of this pyramid?
2. What is the slant height if the height is doubled?
3. How does the total surface area change if the base is halved?
4. What is the relation between the base area and lateral area?
5. Can you derive the surface area formula for a regular pyramid?

### Tip:
For any right square pyramid, always use the Pythagorean theorem to relate slant height and vertical height through the half of the base length.

A right square pyramid has a total surface area of 36,864 square inches, and the combined surface area of the four lateral faces is 20,480 square inches. What is the height, in inches, of this pyramid?

Learn how to determine the height of a right square pyramid with a total surface area of 36,864 square inches and a lateral surface area of 20,480 square inches. This step-by-step solution covers geometry concepts and uses the Pythagorean theorem. Suitable for Grades 9-12.

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