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:
10 = (x+2y)+1/2 [2π(x/2)]
[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.]
10 = (x+2y)+(πx/2)
From here,
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:
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:
To find the critical point, we need to equate equation (3) to zero.
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:
so the function A is maximum at x = 20/(4+π).
Now substituting x = 20/(4+π) in equation (1):
Hence the given window with maximum area has breadth, x = 20/(4+π) and height, y = 10/(4+π).
