The area of a circle which subtends an angle of theta radians at the centre of the circle is 2/3m^2. If the length of the arc of the sector is 2m, Find theta
Let r be the radius of the circle. The area of the sector with angle θ is given by:
Area = (1/2) * r^2 * θ
According to the problem, this area is (2/3)m^2:
(1/2) * r^2 * θ = (2/3)m^2
The length of the arc is given by the formula:
Arc length = r * θ
According to the problem, this length is 2m:
r * θ = 2m
We have a system of two equations with two unknowns, r and θ. We can solve for one of the variables in terms of the other.
From the equation for the arc length, we have:
θ = 2m / r
Now we can substitute this into the equation for the area:
(1/2) * r^2 * (2m / r) = (2/3)m^2
Simplify the equation:
m * r = (2/3)m^2
Divide by m (assuming m ≠ 0):
r = (2/3)m
Now substitute this back into the equation for θ:
θ = 2m / ((2/3)m)
θ = 2m * (3/2m)
θ = 3 radians
So the angle θ is 3 radians.