Q. 94.0( 1 Vote )
Given the perimet
• Rectangle with given perimeter.
Let us consider,
• ‘p’ as the fixed perimeter of the rectangle.
• ‘x’ and ‘y’ be the sides of the given rectangle.
• Diagonal of the rectangle, . (using the hypotenuse formula)
Now as consider the perimeter of the rectangle,
p = 2(x +y)
p = 2x + 2y
Consider the diagonal of the rectangle,
Substituting (1) in the diagonal of the rectangle,
[squaring both sides]
For finding the maximum/ minimum of given function, we can find it by differentiating it with x and then equating it to zero. This is because if the function f(x) has a maximum/minimum at a point c then f’(c) = 0.
Differentiating the equation (2) with respect to x:
= 2x - p + 2x
To find the critical point, we need to equate equation (3) to zero.
4x –p = 0
4x = p
Now to check if this critical point will determine the minimum diagonal, we need to check with second differential which needs to be positive.
Consider differentiating the equation (3) with x:
= 4 + 0
[Since and ]
Now, consider the value of
As , so the function Z is minimum at .
Now substituting in equation (1):
As the sides of the taken rectangle are equal, we can clearly say that a rectangle with minimum diagonal which has a given perimeter is a square.
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