+84 1)What is the length of a 100 degree arc in a circle of radius 9?
2)A circle is divided into 4 sectors, the ratio of whose areas is 1 : 2 : 3 : 4. What is the central angle of the largest sector in degrees?
3)A sector of a circle has a central angle of 80 degrees. If the area of the sector is 32pi, what is the radius of the circle?
4)Sixteen times the reciprocal of the number of inches in the circumference of a circle equals the number of inches in the diameter of the circle. What is the number of square inches in the area of the circle?
5)A rectangular building is 4 meters by 6 meters. A dog is tied to a rope that is 10 meters long, and the other end is tied to the midpoint of one of the long sides of the building. Find the total area of the region that the dog can reach (not including the inside of the building), in square meters.
+128407 1)What is the length of a 100 degree arc in a circle of radius 9?
We have that
Arc Length = 2 pi radius * ( 100/ 360) = 2 pi * 9 * (10/36) = 18 pi * (5/18) =
5 pi units
+128407 3)A sector of a circle has a central angle of 80 degrees. If the area of the sector is 32pi, what is the radius of the circle?
Area = pi * radius^2 ( 80 / 360) =
32 pi = pi * radius^2 ( 2/9) divide out pi
32 = (2/9) * radius^2 multiply both sides by 9/2
32 (9/2) = r^2
144 = r^2 take the positive root of both sides
12 = r
+128407 4)Sixteen times the reciprocal of the number of inches in the circumference of a circle equals the number of inches in the diameter of the circle. What is the number of square inches in the area of the circle?
I like this one.....
Call the number of inches on the circumference, I
So
16/ I = D = 2r ⇒ 16/ I = 2r ⇒ I = 16/2r ⇒ I = 8/r
So.......
Circumference in inches = 2 * pi * r
8/r = 2 * pi * r multiply both sides by r
8 = 2 * pi * r^2 divide both sides by 2 pi
8 / [ 2pi ] = r^2
4/pi = r^2
So....the area is
pi (r^2) =
pi ( 4/pi) = 4 units^2
+128407 5)A rectangular building is 4 meters by 6 meters. A dog is tied to a rope that is 10 meters long, and the other end is tied to the midpoint of one of the long sides of the building. Find the total area of the region that the dog can reach (not including the inside of the building), in square meters.
The dog will be able to reach the area of a semi-circle with a radius of 10 m as well as two quarter circles with a radius of (10 - 3) m = 7m
So.....the total area is
(1/2 ) pi * 10^2 + (1/4) pi (7)^2 + (1/4) pi (7)^2 =
(1/2) * (10)^2 + (1/2) pi * (7)^2 =
(1/2) pi * ( 10^2 + 7^2) =
(1/2) pi ( 100 + 49) =
(1/2) pi (149) =
(149/2) pi ≈ 234.04 m^2
