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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:
Input : r = 4
Output : 16
Input : r = 5
Output :25
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/?2
This 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
C++
<?php
function rectanglearea($r)
{
if ($r < 0)
return -1;
$a = $r * $r;
return $a;
}
$r = 5;
echo rectanglearea($r)."\n";
?>
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Javascript
<script>
function rectanglearea(r)
{
if (r < 0)
return -1;
var a = r * r;
return a;
}
var r = 5;
document.write(parseInt(rectanglearea(r)));
</script>
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OUTPUT :
25
Time Complexity: O(1)
Auxiliary Space: O(1)
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