38.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 ?

Question

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 the height of the small cone be h.

Let the volume of the frustum be V.

Let the painted area of the top part be A1.

Let the painted area of the bottom part be A2.

Volume of the Frustum

The volume of the frustum can be calculated using the formula:

V = (1/3)πh(R^2 + Rr + r^2) where R is the radius of the base of the large cone, r is the radius of the base of the small cone, and h is the height of the frustum.

Here, R = 3 and r = ? (unknown).

Also, the height of the frustum is 4 - h.

Therefore, V = (1/3)π(4 - h)(9 + 3? + ?^2)

Volume of the Small Cone

The volume of the small cone can be calculated using the formula:

V1 = (1/3)π(?^2)h

Painted Area of the Top Part

The painted area of the top part can be calculated using the formula:

A1 = π?l where l is the slant height of the small cone.

Here, l = √(?^2 + h^2).

Painted Area of the Bottom Part

The painted area of the bottom part can be calculated using the formula:

A2 = πRl where l is the slant height of the large cone.

Here, l = √(9 + h^2).

Equating the Ratios

According to the problem statement, the ratio of the volume of the small cone to the volume of the frustum is equal to the ratio of the painted area of the top part to the painted area of the bottom part.

Therefore, we get:

V1/V = A1/A2

Substituting the formulas we derived earlier, we get:

(1/3)π(?^2)h/[(1/3)π(4 - h)(9 + 3? + ?^2)] = π?l/πRl

Canceling out the common terms and simplifying, we get:

h(R^2 + Rr + r^2) = 4?l

Final Calculation

Substituting the values of R and l, we get:

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