Volumen mínimo de cono que se puede circunscribir alrededor de una esfera de radio R

Dada una esfera de radio R, la tarea es encontrar el volumen mínimo del cono que se puede circunscribir alrededor de ella.

Ejemplos:

Input: R = 10 
Output: Volume of cone = 8373.33 
Explanation: 
Radius of cone = 14.14 and Height of cone = 40,
Volume of cone = So, volume = 8373.33Input: R = 4 Output: Volume of cone = 535.89

Planteamiento:
hemos dado una esfera de radio R inscrita en Cono. Necesitamos encontrar el radio y la altura del cono para encontrar el volumen del cono.

En el triángulo AOE y ALC calcule sin(X), es decir, para el triángulo AOE $sen(X) = (R/HR)$ y para el triángulo ALC $sin(X) = (r/\sqrt{r^2 + H^2})$
Ahora, al igualar ambos obtenemos $H = ( (-2 (r^2) R) / (R^2 - r^2))$
Inserte el valor de H en Volumen, es decir, $V = (1/3) * (3.14) * (r^2) * (H)$ y para que el volumen sea mínimo $d(V)/dr = 0$ .
De la ecuación anterior obtenemos $r = \sqrt{2} * R$ y poniendo este valor en H obtenemos $H = 4*R$
Por lo tanto, aplicando la fórmula de volumen de cono y poniendo $r = \sqrt{2} * R$ y $H = 4 * R$ obtenemos el resultado deseado.

C++

// C++ program to find the minimum
// volume of the cone that can be 
// circumscribed about a sphere
// of radius R
#include<bits/stdc++.h>
using namespace std;
 
// Function to find the volume
// of the cone
float Volume_of_cone(float R)
{
     
    // r = radius of cone
    // h = height of cone
    // Volume of cone = (1 / 3) * (3.14) * (r*r) * (h)
    // we get radius of cone from the derivation
    // is root(2) times multiple of R
    // we get height of cone from the derivation
    // is 4 times multiple of R
    float V = (1 / 3.0) * (3.14) * (2 * ( R * R ) ) * (4 * R);
     
    return V;
}
     
// Driver code
int main()
{
    float R = 10.0;
    cout << Volume_of_cone(R);
}
     
// This code is contributed by Samarth

Java

// Java program to find the minimum
// volume of the cone that can be
// circumscribed about a sphere
// of radius R
import java.util.*;
 
class GFG{
 
// Function to find the volume
// of the cone
static double Volume_of_cone(double R)
{
     
    // r = radius of cone
    // h = height of cone
    // Volume of cone = (1 / 3) * (3.14) * (r*r) * (h)
    // we get radius of cone from the derivation
    // is root(2) times multiple of R
    // we get height of cone from the derivation
    // is 4 times multiple of R
    double V = (double)((1 / 3.0) * (3.14) * (2 * (R * R)) *
                                                  (4 * R));
    return V;
}
     
// Driver code
public static void main(String[] args)
{
    double R = 10.0;
    System.out.print(Volume_of_cone(R));
}
}
 
// This code is contributed by sapnasingh4991

Python3

# Python3 program to find the minimum
# Volume of the cone that can be circumscribed
# about a sphere of radius R
 
import math
 
# Function to find the volume
# of the cone
 
def Volume_of_cone(R):
 
    # r = radius of cone
    # h = height of cone
    # Volume of cone = (1 / 3) * (3.14) * (r**2) * (h)
    # we get radius of cone from the derivation
    # is root(2) times multiple of R
    # we get height of cone from the derivation
    # is 4 times multiple of R
     
    V = (1 / 3) * (3.14) * (2 * ( R**2 ) ) * (4 * R)
     
    return V
     
 
# Driver code
if __name__ == "__main__":
     
    R = 10
     
    print(Volume_of_cone(R))

C#

// C# program to find the minimum
// volume of the cone that can be
// circumscribed about a sphere
// of radius R
using System;
class GFG{
 
// Function to find the volume
// of the cone
static double Volume_of_cone(double R)
{
     
    // r = radius of cone
    // h = height of cone
    // Volume of cone = (1 / 3) * (3.14) * (r*r) * (h)
    // we get radius of cone from the derivation
    // is root(2) times multiple of R
    // we get height of cone from the derivation
    // is 4 times multiple of R
    double V = (double)((1 / 3.0) * (3.14) *
                    (2 * (R * R)) * (4 * R));
    return V;
}
     
// Driver code
public static void Main()
{
    double R = 10.0;
    Console.Write(Volume_of_cone(R));
}
}
 
// This code is contributed by Nidhi_biet

Javascript

<script>
// Javascript program to find the minimum
// volume of the cone that can be
// circumscribed about a sphere
// of radius R
 
    // Function to find the volume
    // of the cone
    function Volume_of_cone( R)
    {
 
        // r = radius of cone
        // h = height of cone
        // Volume of cone = (1 / 3) * (3.14) * (r*r) * (h)
        // we get radius of cone from the derivation
        // is root(2) times multiple of R
        // we get height of cone from the derivation
        // is 4 times multiple of R
        let V =  ((1 / 3.0) * (3.14) * (2 * (R * R)) * (4 * R));
        return V;
    }
 
    // Driver code
    let R = 10.0;
    document.write(Volume_of_cone(R));
 
// This code is contributed by 29AjayKumar
</script>

Producción:

8373.333333333332

Complejidad temporal : O(1) desde la realización de operaciones constantes

Publicación traducida automáticamente

Artículo escrito por virusbuddha y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA

C++

Java

Python3

C#

Javascript

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