Find the rectangle of largest area that can be inscribed in a semicircle of diameter 147, assuming that one side of the rectangle lies on the diameter of the semicircle.
The largest possible area is ______?
2007-11-16 04:30:00 · 3 answers · asked by Hinal P 1 in Science & Mathematics ➔ Mathematics
Rectangle of Area 5397.605 Cm^2
2007-11-16 04:44:31 · answer #1 · answered by Murtaza 6 · 0⤊ 0⤋
Draw a picture. Let the radius of the semicircle be r. The two of the corners of teh rectangle tocuh the curve part of the semicircle and are a distance r away from the center. Now you have a right triangle with r as the hypotoneus, the width of the rectangle, w as one leg, and 1/2 the length (L/2) as the other leg. So you can use:
r^2 = (L/2)^2+w^2 to relate all three variables. Now you know r - it is 1/2 the diameter.
Now the area of the rectangle is A = Lw or
A = w*sqrt(4r^2-w^2) using the above relationship. To maximize the area, take teh first derivative with respect to w, set to zero and solve for w.
dA/dw = 0 =sqrt(4r^2-w^2) -w^2/sqrt(4r^2-w^2)
0 = 4r^2 -w^2-w^2 = 2r^2 - w^2 ---> w =sqrt(2)*r
w = 103.944
L = sqrt(4r^2-w^2) = 103.944
So it is a square.
2007-11-16 12:44:18 · answer #2 · answered by nyphdinmd 7 · 0⤊ 0⤋
the square!!
r = radius og circle
d = diameter of circle
d = 147 = r / 2 = 147 / 2 = 73.5 one side
area = 73.5 one side ^2
area = 5402.25
2007-11-16 12:36:29 · answer #3 · answered by Anonymous · 0⤊ 0⤋