SOLUTION: The slant height of a right circular cone is 2 ft. At what distance from the vertex must the slant height be cut by a plane parallel to the base, in order that the lateral surface

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Question 1198559: The slant height of a right circular cone is 2 ft. At what distance from the vertex must the slant height be cut by a plane parallel to the base, in order that the lateral surface may be divided into two equal areas?
Answer by onyulee(41) About Me  (Show Source):
You can put this solution on YOUR website!
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.**