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Given a semicircle of radius r, we have to find the largest rectangle that can be inscribed in the semicircle, with base lying on the diameter.
Examples:
Let r be the radius of the semicircle, x one half of the base of the rectangle, and y the height of the rectangle. We want to maximize the area, A = 2xy. So from the diagram we have,
y = √(r^2 – x^2)
So, A = 2*x*(√(r^2 – x^2)), or dA/dx = 2*√(r^2 – x^2) -2*x^2/√(r^2 – x^2) Setting this derivative equal to 0 and solving for x,
dA/dx = 0
or, 2*√(r^2 – x^2) – 2*x^2/√(r^2 – x^2) = 0
2r^2 – 4x^2 = 0
x = r/√2This is the maximum of the area as,
dA/dx > 0 when x > r/√2
and, dA/dx < 0 when x > r/√2
Since y =√(r^2 – x^2) we then have
y = r/√2
Thus, the base of the rectangle has length = r/√2 and its height has length √2*r/2.
So, Area, A=r^2
#include <bits/stdc++.h> using namespace std; float rectanglearea(float r) { if (r < 0) return -1; float a = r * r; return a; } int main() { float r = 5; cout << rectanglearea(r) << endl; return 0; } |
class GFG { static float rectanglearea(float r) { if (r < 0) return -1; float a = r * r; return a; } public static void main(String[] args) { float r = 5; System.out.println((int)rectanglearea(r)); } } |
def rectanglearea(r) : if r < 0 : return -1 a = r * r return a if __name__ == "__main__" : r = 5 print(rectanglearea(r)) |
using System; class GFG { static float rectanglearea(float r) { if (r < 0) return -1; float a = r * r; return a; } public static void Main() { float r = 5; Console.Write((int)rectanglearea(r)); } } |
<?php function rectanglearea($r) { if ($r < 0) return -1; $a = $r * $r; return $a; } $r = 5; echo rectanglearea($r)."\n"; ?> |
<script> function rectanglearea(r) { if (r < 0) return -1; var a = r * r; return a; } var r = 5; document.write(parseInt(rectanglearea(r))); </script> |
OUTPUT :
25Time Complexity: O(1)
Auxiliary Space: O(1)
Find the area of the largest rectangle that can be inscribed inside a semi circle with a radius of 10 units. Place the length of the rectangle along the diameter. How do you figure this out? I'm tihnking you have to use trig, but what's the formula for it? I forget.
Thanks for the help + time
Find the area of the largest rectangle that can be inscribed inside a semi circle with a radius of 10 units. Place the length of the rectangle along the diameter. How do you figure this out? I'm tihnking you have to use trig, but what's the formula for it? I forget.
Thanks for the help + time
The rectangle of maximum area within the semi-circle is therefore, [2(10)(sqrt2)/2]^2/2 = [20(sqrt2)/2]^2/2.
first, draw a sketch base of the rectangle that lies on the x-axis is \(\displaystyle 2x\) height of the rectangle is \(\displaystyle y = \sqrt{10^2 - x^2}\) {equation for a semicircle of radius 10, remember?} \(\displaystyle A = 2x\sqrt{100 - x^2}\)
find \(\displaystyle \frac{dA}{dx}\) and maximize.