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. The height of the small cone isa)7/3b)5/4c)5/2d)None of theseCorrect answer is option 'C'. Can you explain this answer?

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Answer

A

Answer

Problem Statement: 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. The height of the small cone is

To find: The height of the small cone

Solution:

Let's begin by finding the painted area of the bottom part and the top part of the cone.

Painted area of bottom part:
The painted area of the bottom part is the area of the circular base of the cone, which is given by:

A1 = πr1^2
A1 = π(3)^2
A1 = 9π

Painted area of top part:
The painted area of the top part is the area of the circular base of the small cone, which is given by:

A2 = πr2^2

To find r2, we need to find the radius of the frustum formed by cutting the cone into two parts. The radius of the frustum at the top is the radius of the small cone, which is r2. The radius of the frustum at the bottom is the radius of the cone, which is r1.

To find the radius of the frustum at the top, we can use the similar triangles formed by the frustum and the original cone.

Let x be the height of the small cone. Then, the height of the frustum is 4 - x.

Using the similar triangles, we have:

r1 / (4) = r2 / (4 - x)

Simplifying, we get:

r2 = (4 - x) / 4 * r1
r2 = (4 - x) / 4 * 3
r2 = (4 - x) / 12

Now, we can find the painted area of the top part:

A2 = πr2^2
A2 = π[(4 - x) / 12]^2
A2 = π(16 - 8x + x^2) / 144

Next, we need to find the volume of the bottom part and the top part of the cone.

Volume of bottom part:
The volume of the bottom part is the volume of the frustum, which is given by:

V1 = (1/3)πh(r1^2 + r1r2 + r2^2)

To find h, we can use the similar triangles again:

h / (4 - x) = 4 / 3

Simplifying, we get:

h = 16 / (4 - x) * 3

Now, we can find the volume of the bottom part:

V1 = (1/3)πh(r1^2 + r1r2 + r2^2)
V1 = (1/3)π[(16 / (4 - x) * 3)^2 + (16 / (4 - x) * 3)(3 / 12 * (4 - x)) + (3 / 12 * (4 - x))^2]

Simplifying, we get:

V1 = (1/3)π(16 / (4

Answer

There is a right circular cone with radius 3 units and height h4 units .The surface area 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 of the 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 theheight of the small cone ?

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