Question

if a central angle is very small, there is little difference in the length between an arc and the inscribed chord. see the figure to the right. assume that the mast of a boat is 41.0 ft high. if it subtends an angle of 3°16, how far away is it? (note: when a central angle intercepts an arc, the arc is said to subtend the angle.) the mast is ft away. (do not round until the final answer. then round to the nearest foot as needed.)

Explanation:

Step1: Convert the angle to decimal - degrees

First, convert \(3^{\circ}16'\) to decimal - degrees. Since \(1^{\circ}=60'\), then \(16'=\frac{16}{60}\approx0.2667^{\circ}\). So the angle \(\theta = 3 + 0.2667=3.2667^{\circ}\). Convert this angle to radians. We know that to convert degrees to radians, we use the formula \(\alpha=\theta\times\frac{\pi}{180}\). So \(\alpha = 3.2667\times\frac{\pi}{180}\approx0.057\) radians.

Step2: Use the small - angle approximation

For a small central angle \(\alpha\) (in radians) subtended by an arc of length \(s\) and radius \(r\) (the distance we want to find), when the angle is small, the length of the arc \(s\) is approximately equal to the length of the chord. Here, the height of the mast is the length of the chord (and approximately the length of the arc), \(s = 41.0\) ft. The formula for the length of an arc is \(s = r\alpha\), where \(r\) is the distance from the observer to the mast. Solving for \(r\), we get \(r=\frac{s}{\alpha}\).

Step3: Calculate the distance

Substitute \(s = 41.0\) ft and \(\alpha\approx0.057\) radians into the formula \(r=\frac{s}{\alpha}\). So \(r=\frac{41.0}{0.057}\approx719\) ft.

Answer:

719