There is a right circular cone with base radius 3 units and height 4 units. The surface of this right circular cone is painted. It is then cut into two parts by a plane parallel to the base so that the volume of the top part (the small cone) divided by the volume of the frustum equals the painted area of the top part divided by the painted area of the bottom part. Find the height of the small cone ?

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Answer

Problem Statement

A right circular cone with base radius 3 units and height 4 units is painted. It is then cut into two parts by a plane parallel to the base so that the volume of the top part (the small cone) divided by the volume of the frustum equals the painted area of the top part divided by the painted area of the bottom part. Find the height of the small cone.

Solution

Let's first find the volume and painted area of the whole cone before it is cut into two parts.

Volume of the cone

The volume of a cone is given by the formula:

V = (1/3)πr²h

Substituting r = 3 and h = 4, we get:

V = (1/3)π(3)²(4) = 12π

Painted area of the cone

The lateral surface area of a cone is given by the formula:

A = πrl

where r is the base radius, l is the slant height, and:

l = √(r² + h²)

Substituting r = 3 and h = 4, we get:

l = √(3² + 4²) = 5

Substituting r = 3, l = 5, and π ≈ 3.14, we get:

A = π(3)(5) ≈ 47.1

Volume and painted area of the frustum

When the cone is cut into two parts, we get a frustum and a smaller cone. Let's find the volume and painted area of the frustum first.

The volume of a frustum is given by the formula:

V = (1/3)πh(h₁² + h₂² + h₁h₂)

where h is the height of the frustum, h₁ and h₂ are the heights of the top and bottom bases, respectively.

Since the plane that cuts the cone is parallel to the base, the top and bottom bases of the frustum are similar to the original base of the cone. Thus, the ratio of their radii is the same as the ratio of their heights:

r₁/r₂ = h₁/h₂

Substituting r₁ = 1 and r₂ = 3, we get:

1/3 = h

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