Question

circular railroad curves are designated by the degree of curvature, the central angle subtended by a chord of 100 ft. suppose a portion of track has a curvature of 20.0°. find (a) the radius of the curve, (b) the length of the arc determined by the 100 - ft chord, and (c) the area of the portion of the circle bounded by the arc and the 100 - ft chord. (a) the radius of the curve is □ ft. (round to the nearest integer as needed.)

Explanation:

Step1: Recall the relationship between chord - length, radius and central - angle

Let the radius of the circle be $r$, the length of the chord be $c = 100$ ft and the central - angle be $\theta=29^{\circ}$. First, convert the angle to radians. $\theta = 29^{\circ}\times\frac{\pi}{180}\approx0.506$ radians. We know the formula $c = 2r\sin(\frac{\theta}{2})$.

Step2: Solve for the radius $r$

From $c = 2r\sin(\frac{\theta}{2})$, we can re - arrange it to get $r=\frac{c}{2\sin(\frac{\theta}{2})}$. Substitute $c = 100$ and $\theta = 29^{\circ}$ (or $\theta\approx0.506$ radians). $\frac{\theta}{2}=14.5^{\circ}\approx0.253$ radians, and $\sin(\frac{\theta}{2})=\sin(14.5^{\circ})\approx0.25$. Then $r=\frac{100}{2\times0.25}=200$ ft.

Answer:

200