Question 1198258
Certainly, let's find the lateral surface area of the square pyramid.

**1. Define Variables**

* Let 's' be the side length of the square base.
* Let 'l' be the length of the lateral edge (l = 3.25s).
* Let 'h' be the height of the pyramid.

**2. Find the Volume of the Pyramid**

The volume of a square pyramid is given by:

* Volume = (1/3) * Base Area * Height 
* 250 = (1/3) * s^2 * h

**3. Find the Height of the Pyramid**

* We know that the lateral edge, base edge, and height form a right triangle. 
* Using the Pythagorean Theorem:
    * h^2 + (s/2)^2 = l^2 
    * h^2 + (s^2)/4 = (3.25s)^2
    * h^2 = (3.25s)^2 - (s^2)/4
    * h^2 = 10.5625s^2 - 0.25s^2
    * h^2 = 10.3125s^2
    * h = √(10.3125s^2) 
    * h = s√10.3125

**4. Substitute 'h' in the Volume Equation**

* 250 = (1/3) * s^2 * (s√10.3125)
* 250 = (√10.3125/3) * s^3
* s^3 = 250 / (√10.3125/3)
* s^3 ≈ 22.81
* s ≈ 2.84 inches

**5. Calculate the Lateral Edge Length**

* l = 3.25s 
* l = 3.25 * 2.84 
* l ≈ 9.21 inches

**6. Calculate the Slant Height**

* Let 'L' be the slant height.
* L^2 = h^2 + (s/2)^2
* L^2 = (s√10.3125)^2 + (s/2)^2
* L^2 = 10.3125s^2 + 0.25s^2
* L^2 = 10.5625s^2
* L = s√10.5625
* L = 2.84 * √10.5625 
* L ≈ 9.31 inches

**7. Calculate the Lateral Surface Area**

* Lateral Surface Area = 4 * (1/2) * Base Edge * Slant Height
* Lateral Surface Area = 4 * (1/2) * s * L
* Lateral Surface Area = 4 * (1/2) * 2.84 * 9.31 
* Lateral Surface Area ≈ 52.89 square inches

**Therefore, the lateral surface area of the square pyramid is approximately 53 square inches.**