Q. 94.0( 1 Vote )

# Given the perimet

Answer :

Given,

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 ----- (1)

Consider the diagonal of the rectangle, Substituting (1) in the diagonal of the rectangle, [squaring both sides] ----- (2)

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:   [Since ]

= 2x - p + 2x ------ (3)

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 ] ------ (4)

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.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation view all courses RELATED QUESTIONS :

If the sum of theMathematics - Board Papers

A metal box with Mathematics - Board Papers

Show that aRD Sharma - Volume 1

Find the local maMathematics - Board Papers

Prove that the seMathematics - Board Papers

Prove that the raMathematics - Board Papers

Prove that the leMathematics - Board Papers