Sure, let's solve this step-by-step:
1. Determine the time flown in hours:
- The pilot flies the first leg for 105 minutes. Since there are 60 minutes in an hour, we convert this time to hours:
[tex]\[
\text{Time for the first leg} = \frac{105}{60} \approx 1.75 \text{ hours}
\][/tex]
- The pilot then flies the second leg for 80 minutes, so we convert this time to hours as well:
[tex]\[
\text{Time for the second leg} = \frac{80}{60} \approx 1.3333 \text{ hours}
\][/tex]
2. Calculate the distances for each leg:
- The plane's speed is 660 miles per hour. We can use this to calculate the distance flown during each leg using [tex]\( \text{distance} = \text{speed} \times \text{time} \)[/tex]:
[tex]\[
\text{Distance for the first leg} = 660 \times 1.75 = 1,155 \text{ miles}
\][/tex]
[tex]\[
\text{Distance for the second leg} = 660 \times 1.3333 \approx 880 \text{ miles}
\][/tex]
3. Calculate the straight-line distance from the starting position to the final position:
- The pilot changes course by [tex]\( 15^\circ \)[/tex] to the right of her original direction. We use the law of cosines to find the straight-line distance from the start to the end point of her journey. The law of cosines states:
[tex]\[
c = \sqrt{a^2 + b^2 - 2ab \cos(\theta)}
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the distances of the two legs, and [tex]\( \theta \)[/tex] is the angle between them in degrees (which we convert to radians for calculation):
- Converting [tex]\( 15^\circ \)[/tex] to radians (since most trigonometric functions use radians):
[tex]\[
15^\circ = 15 \times \frac{\pi}{180} \approx 0.2618 \text{ radians}
\][/tex]
- Applying the law of cosines:
[tex]\[
\text{Distance from start} = \sqrt{1155^2 + 880^2 - 2 \times 1155 \times 880 \times \cos(0.2618)}
\][/tex]
- After evaluating the above expression, we find:
[tex]\[
\text{Distance from start} \approx 380.645478522446 \text{ miles}
\][/tex]
4. Rounding the result to the nearest mile:
[tex]\[
\text{Rounded distance} = 381 \text{ miles}
\][/tex]
Thus, the pilot is approximately 381 miles from her starting position after making the course correction.
