Answer :

Given,


Window is in the form of a rectangle which has a semicircle mounted on it.


Total Perimeter of the window is 10 metres.


The total area of the window is maximum.



Let us consider,


The breadth and height of the rectangle be ‘x’ and ‘y’.


The radius of the semicircle will be half of the base of the rectangle.


Given Perimeter of the window is 10 meters:



[as the perimeter of the window will be equal to one side (x) less to the perimeter of rectangle and the perimeter of the semicircle.]



From here,



----- (1)


Now consider the area of the window,


Area of the window = area of the semicircle + area of the rectangle



Substituting (1) in the area equation:





------ (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 ]



------ (3)


To find the critical point, we need to equate equation (3) to zero.





x (4 +π) = 20



Now to check if this critical point will determine the maximum area of the window, we need to check with second differential which needs to be negative.


Consider differentiating the equation (3) with x:




[Since ]


------ (4)


As , so the function A is maximum at
.


Now substituting in equation (1):





Hence the given window with maximum area has breadth, and height, .


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
caricature
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