Respuesta :
Answer:
132.233 ft2
Step-by-step explanation:
Let's call the width of the rectangle 'w' and the length 'x'. So the area of the semicircle is:
[tex]A_1 = \pi*radius^2/2[/tex]
[tex]A_1 = \pi*(w/2)^2/2[/tex]
[tex]A_1 = \pi/8*w^2[/tex]
And the area of the rectangle is:
[tex]A_2 = w*x[/tex]
If the perimeter of the window is 41 feet, we have:
[tex]Perimeter = length + 2*width + \pi*radius[/tex]
[tex]41 = x + 2*w + \pi*w/2[/tex]
[tex]x = 41 - w(2 + \pi/2)[/tex]
Now, the equation for the total area of the window is:
[tex]A = A_1 + A_2 = \pi/8*w^2 + w*x[/tex]
[tex]A = \pi/8*w^2 + w*(41 - w(2 + \pi/2))[/tex]
[tex]A = (\pi/8-2 - \pi/2)*w^2 + 41w = -3.1781w^2 + 41w[/tex]
To find the maximum area, we can find the x-coordinate of the vertex of the quadratic equation:
[tex]x\_vertex = -b / 2a = -41 / (-3.1781*2) = 6.45[/tex]
So the width that gives us the maximum area of the window is 6.45 feet, and the area will be:
[tex]A = -3.1781w^2 + 41w = -3.1781*(6.45)^2 + 41*6.45 = 132.233\ ft^2[/tex]
