Question

if a central angle is very small, there is little difference in length between an arc and the inscribed chord. see the figure. approximate the following length by finding the arc length. (note: when a central angle intercepts an arc, the arc is said to subtend the angle.) the ends of the railroad track in the desert are 7.5 km away from the vertex of the subtended angle. a train on the track subtends (horizontally) an angle of 4°20′. find the approximate length of the train. the length of the train is about □ km. (simplify your answer. type an integer or a decimal rounded to two decimal places as needed.)

Explanation:

Step1: Convert the angle to radians

First, convert $4^{\circ}20'$ to degrees. Since $20'=\frac{20}{60}=\frac{1}{3}\approx0.333^{\circ}$, then $4^{\circ}20' = 4+\frac{1}{3}=\frac{13}{3}\approx4.333^{\circ}$. To convert degrees to radians, use the formula $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. So $\theta=\frac{13}{3}\times\frac{\pi}{180}=\frac{13\pi}{540}\text{ radians}$.

Step2: Use the arc - length formula

The arc - length formula is $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians. Here, $r = 7.5$ km and $\theta=\frac{13\pi}{540}$. Then $s=7.5\times\frac{13\pi}{540}=\frac{7.5\times13\pi}{540}=\frac{97.5\pi}{540}$.

Step3: Calculate the value

$s=\frac{97.5\pi}{540}\approx\frac{97.5\times3.14159}{540}\approx\frac{306.305}{540}\approx0.57$ km.

Answer:

$0.57$