Question

if \\( \theta = 33 ^ { \circ } \\), find the distance between two cities, a and b, to the nearest mile. the radius of the earth is approximately 4000 miles. the distance between the two cities, a and b, is approximately \\( \square \\) miles. (round to the nearest whole number as needed.)

Explanation:

Step1: Recall the arc length formula

The formula for the length of an arc \( s \) of a circle with radius \( r \) and central angle \( \theta \) (in degrees) is \( s = 2\pi r\times\frac{\theta}{360^\circ} \), which can also be written as \( s = r\theta\times\frac{\pi}{180^\circ} \) (converting degrees to radians). Here, \( r = 4000 \) miles and \( \theta = 33^\circ \).

Step2: Substitute the values into the formula

Substitute \( r = 4000 \) and \( \theta = 33^\circ \) into the arc length formula \( s = 4000\times33^\circ\times\frac{\pi}{180^\circ} \).

First, simplify the expression:
\( s = 4000\times\frac{33\pi}{180} \)
\( s = \frac{4000\times33\pi}{180} \)
Simplify the fraction \( \frac{33}{180}=\frac{11}{60} \), so:
\( s = 4000\times\frac{11\pi}{60} \)
\( s=\frac{4000\times11\pi}{60}=\frac{44000\pi}{60}=\frac{2200\pi}{3}\approx\frac{2200\times 3.1416}{3} \)

Step3: Calculate the numerical value

Calculate \( 2200\times3.1416 = 6911.52 \), then divide by 3:
\( \frac{6911.52}{3}\approx2303.84 \approx 2304 \) (rounded to the nearest mile)

Answer:

2304