Respuesta :
Answer:
Step-by-step explanation:
Let a is the width of the window and diameter of the semicircle and let h be height of the rectangular portion of the window
:
Perimeter:
2h + a + .5x*pi = 21
2h + 2.57a = 21
2h = 21 - 2.57a
h = (10.5-1.285a)
:
What would be the window with the greatest area;
Area = semicircle + rectangle
Radius = .5a
A = (.5*pi*(.5a)^2) + h*a
Replace h with (10.5-1.285a
A = (1.57*.25a^2) + x(10.5-1.285a)
A = .3927a^2 - 1.285a^2 + 10.5a
A = -.8923a^2 + 10.5a
Find the max area by finding the axis of symmetry; x = -b/(2a)
a = 5.88 meter is the width with the greatest area
:
Find the max area
A = -.8923(5.88^2) + 10.5(5.88)
A = -.8923(5.88^2) + 10.5(5.88)
A = -30.85 + 61.74
A = 30.89 sq/ft is max area
We want to find the dimensions of the window that maximize the area of the window.
The dimensions that maximize the area are:
Length = 6ft
width = 6ft
Then the maximum area is A = 36ft^2
We know that the window is a rectangle, and we know that for a rectangle of width W and length L, the area is given by:
A = L*W
And the perimeter is given by:
P = 2*(L + W).
Here we know that the perimeter is equal to 24ft, then we can write:
24ft = 2*(L + W)
24ft/2 = L + W
12 ft = L + W
Now we can isolate one of the dimensions to get:
W = 12ft - L
Now we can replace this on the area equation to get:
A = L*(12ft - L) = 12ft*L - L^2
Now we want to maximize this, notice that this is a quadratic equation with a negative leading coefficient, meaning that the maximum is at the vertex, and the vertex of the quadratic equation is:
L = - 12ft/(2*(-1)) = 6ft.
And: W = 12ft - L = 12ft - ft = 6ft.
Then the length and the width are equal to 6ft, meaning that this is a square.
The maximum area is:
A = 6ft*6ft = 36ft^2
If you want to learn more, you can read:
https://brainly.com/question/3672366
