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A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total height of the solid is 3 times the volume of the cone, then the height of the cylinder is?
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A solid consists of a circular cylinder surmounted by a right circular...
Total Height and Volume Relationship
The solid consists of a circular cylinder and a right circular cone. Let's analyze the relationship between the height of the cone and the total height of the solid.
Definitions and Variables
- Let the height of the cone be h.
- Let the radius of the cone and cylinder be r.
- Let the height of the cylinder be H.
Volume of the Cone
The volume (V) of the cone is given by:
- V = (1/3) * π * r^2 * h
Total Height of the Solid
The total height of the solid (T) is:
- T = H + h
According to the problem, the total height of the solid is three times the volume of the cone:
- H + h = 3 * V
Substituting the volume of the cone:
- H + h = 3 * (1/3) * π * r^2 * h
- H + h = π * r^2 * h
Finding the Height of the Cylinder
To find H, rearrange the equation:
- H = π * r^2 * h - h
- H = h(π * r^2 - 1)
Conclusion
The height of the cylinder (H) is expressed in terms of the height of the cone (h) and the radius (r). Depending on the value of the radius, you can calculate H accordingly. The relationship highlights how the geometric properties affect the dimensions of the solid.
The solid consists of a circular cylinder and a right circular cone. Let's analyze the relationship between the height of the cone and the total height of the solid.
Definitions and Variables
- Let the height of the cone be h.
- Let the radius of the cone and cylinder be r.
- Let the height of the cylinder be H.
Volume of the Cone
The volume (V) of the cone is given by:
- V = (1/3) * π * r^2 * h
Total Height of the Solid
The total height of the solid (T) is:
- T = H + h
According to the problem, the total height of the solid is three times the volume of the cone:
- H + h = 3 * V
Substituting the volume of the cone:
- H + h = 3 * (1/3) * π * r^2 * h
- H + h = π * r^2 * h
Finding the Height of the Cylinder
To find H, rearrange the equation:
- H = π * r^2 * h - h
- H = h(π * r^2 - 1)
Conclusion
The height of the cylinder (H) is expressed in terms of the height of the cone (h) and the radius (r). Depending on the value of the radius, you can calculate H accordingly. The relationship highlights how the geometric properties affect the dimensions of the solid.
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A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total height of the solid is 3 times the volume of the cone, then the height of the cylinder is?
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A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total height of the solid is 3 times the volume of the cone, then the height of the cylinder is? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total height of the solid is 3 times the volume of the cone, then the height of the cylinder is? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total height of the solid is 3 times the volume of the cone, then the height of the cylinder is?.
A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total height of the solid is 3 times the volume of the cone, then the height of the cylinder is? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total height of the solid is 3 times the volume of the cone, then the height of the cylinder is? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total height of the solid is 3 times the volume of the cone, then the height of the cylinder is?.
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