Question

Как относятся объемы равностороннего конуса и равностороннего цилиндра если их полные поверхности равновелики.

Answer 1

Sп.к.=πR²+πRL=πR²+2πR²=3πR²
L=2R
Sп.ц.=2πrh+2πr²=4πr²+2πr²=6πr²
h=2r
3πR²=6πr²⇒R²=2r²⇒R=r√2
Vк=1/3*πR²H=√3πR³/3=2√6πr³/3
H=R√3
Vц=πr²h=2πr³
Vк/Vц=2√6r³/3 :2πr³=√6/3=√2/√3

Answer 2

Relationship Between the Volumes of an Equilateral Cone and an Equilateral Cylinder

When the full surfaces of an equilateral cone and an equilateral cylinder are equal, their volumes have a specific relationship. Let's explore this in detail.

Equilateral Cone and Equilateral Cylinder: - An equilateral cone is a cone with a circular base and all its lateral edges being equal in length. - An equilateral cylinder is a cylinder with a circular base and all its lateral edges being equal in length.

Relationship Between Their Volumes: - The relationship between the volumes of an equilateral cone and an equilateral cylinder when their full surfaces are equal is as follows: - The volume of the equilateral cone is one-third the volume of the equilateral cylinder with the same base and height.

Explanation: - The volume of a cone is given by the formula \(V = \frac{1}{3} \pi r^2 h\), where \(r\) is the radius of the base and \(h\) is the height. - The volume of a cylinder is given by the formula \(V = \pi r^2 h\), where \(r\) is the radius of the base and \(h\) is the height. - When the full surfaces of the equilateral cone and the equilateral cylinder are equal, the volume of the cone is one-third the volume of the cylinder with the same base and height.

This relationship highlights the proportional difference in volumes between an equilateral cone and an equilateral cylinder when their full surfaces are equal.

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