In a circle, the ratio of the length of an arc intercepted by an angle to the length of the radius of the circle is equal to the radian measure of the angle.
So if the angle intercepts an arc of length 41.6 in a circle with radius r, then the radian measure of the angle is:
θ = arc length / radius
θ = 41.6 / r
However, we don't know the value of the radius, so we need another equation to solve for it. We can use the fact that the total angle in a circle is 2π radians (or 360 degrees). So if the angle intercepts some portion of the circle, the remaining angle in the circle is:
2π - θ
Since this remaining angle intercepts an arc of length equal to the circumference of the circle minus the arc length intercepted by the original angle, we have:
2π - θ = (2πr - 41.6) / r
Now we have two equations with two unknowns (θ and r), which we can solve simultaneously. Rearranging the second equation, we get:
2πr - 41.6 = (2π - θ) r
r = 41.6 / (2π - θ)
Substituting this expression for r into the first equation, we get:
θ = 41.6 / (41.6 / (2π - θ))
θ = 2π - θ
Solving for θ, we get:
2θ = 2π
θ = π
Therefore, the angle in radians to the nearest tenth is approximately 3.1.