The area of the sector is
The length of the curve is:
Using the two equations, solve for r and theta.
BTW, use proper grouping symbols. The way you have them written would suggest . I assume you mean what I have above?.
im not sure how i can draw the specific image i have for the problem to be solved, but i'll try to explain it and then maybe someone can help me out with this. so there is a circle with radius r and center O. The points A, B, and C are on the circle and AOC = the angle θ, situated across the arc ABC. The area of the sector OABC is 4/3π and the length of the arc ABC is 2/3π.
Find the value of r and θ.
Thanks for your help!
Area of Sector OABC = [(2pi -theta)/(2pi)]*pi(r^2) = (4/3)pi
[(2pi)/(2pi) -theta/(2pi)]*pi(r^2) = 4pi/3
[1 -theta/(2pi)](r^2) = 4/3 -----------------(1)
Arc ABC = r(theta) = (2/3)pi
theta = 2pi/(3r) -----------------------(2)
Substitute that into (1),
[1 -(2pi)/(3r)/(2pi)]r^2 = 4/3
[1 -1/(3r)]r^2 = 4/3
r^2 -r/3 = 4/3
Clear the fractions, multiply both sides by 3,
3r^2 -r = 4
3r^2 -r -4 = 0
Factor that, ------- or use the Quadratic Formula
(3r -4)(r +1) = 0
r = 4/3 or -1
Reject r = -1 because there are no negative dimensions.
So, r = 4/3 units -------------------answer.
theta = 2pi/(3r) -------------(2)
theta = 2pi/[3(4/3)]
theta = pi/2 radians -------------------answer.
Hello, miley_22!
Another approach . . .
There is a circle with radius r and center O.
The points A, B, and C are on the circle
and subtends arc
The area of the sector
. . and the length of the arc
Find the value of and .Code:* * * * * A * o * / o B / * / θ * * * - - - - o C * O * * * * * * * * * *
The area of the sector is: . .[1]
The arc length of is: . .[2]
Divide [1] by [2]: .
Substitute into [2]: .