Question 1198559
Certainly, let's find the distance from the vertex where the slant height must be cut to divide the lateral surface of the cone into two equal parts.

**1. Understand the Problem**

* We have a right circular cone.
* The slant height is 2 feet.
* A plane parallel to the base cuts the cone, dividing the lateral surface into two equal parts.
* We need to find the distance of the cut from the vertex.

**2. Let's Define**

* Let 'r' be the radius of the base of the original cone.
* Let 'l' be the slant height of the original cone (l = 2 feet).
* Let 'h' be the height of the original cone.
* Let 'r1' be the radius of the base of the smaller cone formed by the cut.
* Let 'l1' be the slant height of the smaller cone.

**3. Find the Lateral Surface Area of the Original Cone**

* Lateral Surface Area (original) = π * r * l 
* Lateral Surface Area (original) = π * r * 2 
* Lateral Surface Area (original) = 2πr

**4. Find the Lateral Surface Area of the Smaller Cone**

* Lateral Surface Area (smaller) = π * r1 * l1

**5. Divide the Lateral Surface Area Equally**

* Since the plane divides the lateral surface into two equal parts:
    * Lateral Surface Area (smaller) = (1/2) * Lateral Surface Area (original)
    * π * r1 * l1 = (1/2) * 2πr
    * π * r1 * l1 = πr
    * r1 * l1 = r

**6. Relate Radii and Slant Heights**

* In similar cones, the ratio of radii is equal to the ratio of slant heights.
* r1 / r = l1 / l 
* r1 / r = l1 / 2

**7. Substitute and Solve**

* From equation (5): r1 * l1 = r
* Substitute l1 = (r1 * 2) / r 
* r1 * [(r1 * 2) / r] = r
* 2 * r1² = r²
* r1² = (r²)/2
* r1 = (r/√2)

**8. Find the Distance from the Vertex**

* Since r1 / r = l1 / l 
* l1 / l = (r/√2) / r 
* l1 / l = 1/√2
* l1 = l / √2
* l1 = 2 / √2 
* l1 = √2 feet

**Therefore, the slant height must be cut at a distance of √2 feet from the vertex to divide the lateral surface of the cone into two equal parts.**