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Maximize a value for a semicircle of given radius

  • Difficulty Level : Medium
  • Last Updated : 05 May, 2021

We are given a semi circle with radius R. We can take any point on the circumference let it be P.Now, from that point P draw two lines to the two sides of diameter. Let the lines be PQ and PS.
The task is to find the maximum value of expression PS2 + PQ for a given R.
 

semi-circle

Examples : 
 

Input : R = 1 
Output : 4.25  
(4*1^2 + 0.25) = 4.25

Input : R = 2
Output : 16.25 
(4 * 2^2 + 0.25)= 16.25

 

Let F = PS^2 + PQ. We know QS = 2R (Diameter of the semicircle) 
-> We also know that triangle PQS will always be right angled triangle irrespective of the position of point P on the circumference of circle 
 

1.)QS^2 = PQ^2 + PS^2 (Pythagorean Theorem)

2.) Adding and Subtracting PQ on the RHS
     QS^2 = PQ^2 + PS^2 + PQ - PQ

3.) Since QS = 2R
   4*R^2 + PQ - PQ^2 = PS^2 + PQ 
=> 4*R^2 + PQ - PQ^2 = F

4.) Using the concept of maxima and minima 
differentiating F with respect to PQ and equating 
it to 0 to get the point of maxima for F i.e.
   1 - 2 * PQ = 0
 => PQ = 0.5

5.) Now F will be maximum at F = 4*R^2 + 0.25 

 

C++




// C++ program to find
// the maximum value of F
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the
// maximum value of F
double maximumValueOfF (int R)
{
    // using the formula derived for
    // getting the maximum value of F
    return 4 * R * R + 0.25;    
}
     
// Drivers code
int main()
{
    int R = 3;
    printf("%.2f", maximumValueOfF(R));
    return 0;
}

JAVA




// JAVA program to find
// the maximum value of F
import java.io.*;
 
class GFG
{
     
    // Function to find the
    // maximum value of F
    static double maximumValueOfF (int R)
    {
         
        // using the formula derived for
        // getting the maximum value of F
        return 4 * R * R + 0.25;    
    }
     
    // Driver code
    public static void main (String[] args)
    {
        int R = 3;
        System.out.println(maximumValueOfF(R));
    }
         
}
 
// This code is contributed
// by anuj_67.

Python3




# python program to find
# the maximum value of F
 
# Function to find the
# maximum value of F
def maximumValueOfF (R):
     
    # using the formula derived for
    # getting the maximum value of F
    return 4 * R * R + 0.25
 
      
# Drivers code
R = 3
print(maximumValueOfF(R))
 
# This code is contributed by Sam007.

C#




// C# program to find the
// maximum value of F
using System;
 
class GFG
{
     
    // Function to find the
    // maximum value of F
    static double maximumValueOfF (int R)
    {
         
        // using the formula derived for
        // getting the maximum value of F
        return 4 * R * R + 0.25;    
    }
     
    // Driver code
    public static void Main ()
    {
        int R = 3;
        Console.WriteLine(maximumValueOfF(R));
    }
         
}
 
// This code is contributed by Sam007.

PHP




<?php
// PHP program to find the
// maximum value of F
 
// Function to find the
// maximum value of F
function maximumValueOfF ($R)
{
     
    // using the formula derived
    // for getting the maximum
    // value of F
    return 4 * $R * $R + 0.25;
}
     
// Drivers code
$R = 3;
echo maximumValueOfF($R);
 
// This code is contributed
// by anuj_67.
?>

Javascript




<script>
// javascript program to find the
// maximum value of F
 
 // Function to find the
 // maximum value of F
     
 function maximumValueOfF(R) {
           
   // using the formula derived for
   // getting the maximum value of F
         
    return 4 * R * R + 0.25;    
  }
       
  // Driver code
 
  var R = 3;
  document.write(maximumValueOfF(R));
 
// This code is contributed by bunnyram19.
</script>
Output : 
36.25

 


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