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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 A: 7/3 B: 5/4 C: 5/2 D: None of these Please answer it with an explanation!?
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There is a right circular cone with base radius 3 units and height 4 u...
To solve this problem, we need to find the height of the small cone. Let's break down the problem into smaller steps.
Step 1: Calculating the volume and surface area of the original cone
Given that the base radius is 3 units and the height is 4 units, we can calculate the volume and surface area of the original cone using the following formulas:
Volume of a cone = (1/3) * π * r^2 * h
Surface area of a cone = π * r * (r + l)
Where r is the radius and h is the height, and l represents the slant height of the cone.
Step 2: Calculating the volume and surface area of the small cone
Let's assume the height of the small cone is 'x'. We need to find the volume and surface area of this small cone.
Volume of the small cone = (1/3) * π * r^2 * x
Surface area of the small cone = π * r * (r + l')
Here, r is the radius of the small cone, and l' represents the slant height of the small cone.
Step 3: Setting up the equation
The problem states that the volume of the small cone divided by the volume of the frustum (the remaining part) is equal to the painted area of the small cone divided by the painted area of the frustum.
So, we can set up the equation as follows:
(1/3) * π * r^2 * x / [(1/3) * π * (R^2 + R*r + r^2) * (4 - x)] = π * r * (r + l') / [π * (R + r) * l]
Simplifying the equation:
r * x / [(R^2 + R*r + r^2) * (4 - x)] = r * (r + l') / [(R + r) * l]
Step 4: Simplifying the equation
Let's simplify the equation further by canceling out common terms:
x / [(R^2 + R*r + r^2) * (4 - x)] = r + l' / [(R + r) * l]
Step 5: Substituting the values
Now, we can substitute the given values into the equation. The base radius is 3 units, and the height of the original cone is 4 units.
Substituting these values, we get:
x / [(9 + 3r + r^2) * (4 - x)] = r + l' / [(3 + r) * l]
Step 6: Solving the equation
To solve this equation, we need to eliminate the variables. We can do this by using the given information that the plane is parallel to the base.
Since the plane is parallel to the base, the small cone and the frustum have similar triangles.
Using the similar triangles, we can express l' in terms of x and l:
l' / l = x / (4 - x)
Substituting this into the equation, we get:
x / [(9 + 3r + r^2) * (4 - x)] = r + x / (3 + r)
Step
Step 1: Calculating the volume and surface area of the original cone
Given that the base radius is 3 units and the height is 4 units, we can calculate the volume and surface area of the original cone using the following formulas:
Volume of a cone = (1/3) * π * r^2 * h
Surface area of a cone = π * r * (r + l)
Where r is the radius and h is the height, and l represents the slant height of the cone.
Step 2: Calculating the volume and surface area of the small cone
Let's assume the height of the small cone is 'x'. We need to find the volume and surface area of this small cone.
Volume of the small cone = (1/3) * π * r^2 * x
Surface area of the small cone = π * r * (r + l')
Here, r is the radius of the small cone, and l' represents the slant height of the small cone.
Step 3: Setting up the equation
The problem states that the volume of the small cone divided by the volume of the frustum (the remaining part) is equal to the painted area of the small cone divided by the painted area of the frustum.
So, we can set up the equation as follows:
(1/3) * π * r^2 * x / [(1/3) * π * (R^2 + R*r + r^2) * (4 - x)] = π * r * (r + l') / [π * (R + r) * l]
Simplifying the equation:
r * x / [(R^2 + R*r + r^2) * (4 - x)] = r * (r + l') / [(R + r) * l]
Step 4: Simplifying the equation
Let's simplify the equation further by canceling out common terms:
x / [(R^2 + R*r + r^2) * (4 - x)] = r + l' / [(R + r) * l]
Step 5: Substituting the values
Now, we can substitute the given values into the equation. The base radius is 3 units, and the height of the original cone is 4 units.
Substituting these values, we get:
x / [(9 + 3r + r^2) * (4 - x)] = r + l' / [(3 + r) * l]
Step 6: Solving the equation
To solve this equation, we need to eliminate the variables. We can do this by using the given information that the plane is parallel to the base.
Since the plane is parallel to the base, the small cone and the frustum have similar triangles.
Using the similar triangles, we can express l' in terms of x and l:
l' / l = x / (4 - x)
Substituting this into the equation, we get:
x / [(9 + 3r + r^2) * (4 - x)] = r + x / (3 + r)
Step
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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 A: 7/3 B: 5/4 C: 5/2 D: None of these Please answer it with an explanation!?
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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 A: 7/3 B: 5/4 C: 5/2 D: None of these Please answer it with an explanation!? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about 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 A: 7/3 B: 5/4 C: 5/2 D: None of these Please answer it with an explanation!? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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 A: 7/3 B: 5/4 C: 5/2 D: None of these Please answer it with an explanation!?.
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 A: 7/3 B: 5/4 C: 5/2 D: None of these Please answer it with an explanation!? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about 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 A: 7/3 B: 5/4 C: 5/2 D: None of these Please answer it with an explanation!? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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 A: 7/3 B: 5/4 C: 5/2 D: None of these Please answer it with an explanation!?.
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