Question
A U.S. Coast Guard Response Boat leaves Charleston, South Carolina at 1:00 a.m. heading due east at an average speed of 46 knots (nautical miles per hour). At 4:30 a.m., the boat changes course to N 21°
E. At 10:30 a.m. what is the boat's bearing and distance from Charleston, South Carolina? Round all units to the nearest hundredth.
Bearing from Charleston, South Carolina:
(direction)
____ °
(direction)
Distance from Charleston, South Carolina:
______ nautical miles
E. At 10:30 a.m. what is the boat's bearing and distance from Charleston, South Carolina? Round all units to the nearest hundredth.
Bearing from Charleston, South Carolina:
(direction)
____ °
(direction)
Distance from Charleston, South Carolina:
______ nautical miles
Answers
GPT-4o mini
To find the bearing and distance from Charleston to the U.S. Coast Guard Response Boat at 10:30 a.m., we need to break down the boat's journey into two segments:
1. From 1:00 a.m. to 4:30 a.m. (3.5 hours) heading due east.
2. From 4:30 a.m. to 10:30 a.m. (6 hours) heading N 21° E.
**Step 1: Calculate the distance traveled east from 1:00 a.m. to 4:30 a.m.**
The average speed of the boat is 46 knots.
The time from 1:00 a.m. to 4:30 a.m. is 3.5 hours.
\[
\text{Distance} = \text{Speed} \times \text{Time} = 46 \text{ knots} \times 3.5 \text{ hours} = 161 \text{ nautical miles}
\]
**Step 2: Calculate the boat's position at 4:30 a.m.**
At 4:30 a.m., the boat has traveled 161 nautical miles due east. We can represent this position in a coordinate system as:
- Easting: \( 161 \) nautical miles.
- Northing: \( 0 \) nautical miles.
**Step 3: Calculate the distance traveled from 4:30 a.m. to 10:30 a.m. at N 21° E.**
The time from 4:30 a.m. to 10:30 a.m. is 6 hours.
\[
\text{Distance} = \text{Speed} \times \text{Time} = 46 \text{ knots} \times 6 \text{ hours} = 276 \text{ nautical miles}
\]
**Step 4: Find the components of the distance traveled at N 21° E.**
Using trigonometry, we can break down the distance of 276 nautical miles into its eastward and northward components:
\[
\text{East Component} = 276 \times \cos(21^\circ)
\]
\[
\text{North Component} = 276 \times \sin(21^\circ)
\]
Calculating these:
- **East Component:**
\[
\text{East Component} = 276 \times \cos(21^\circ) \approx 276 \times 0.93358 \approx 257.9 \text{ nautical miles}
\]
- **North Component:**
\[
\text{North Component} = 276 \times \sin(21^\circ) \approx 276 \times 0.35837 \approx 98.8 \text{ nautical miles}
\]
**Step 5: Add the two segments together.**
Now we calculate the total east and north distances from Charleston at 10:30 a.m.:
- Total Easting:
\[
\text{Total Easting} = 161 + 257.9 \approx 418.9 \text{ nautical miles}
\]
- Total Northing:
\[
\text{Total Northing} = 0 + 98.8 \approx 98.8 \text{ nautical miles}
\]
**Step 6: Calculate the total distance from Charleston.**
Using the Pythagorean theorem to find the distance from Charleston:
\[
\text{Distance} = \sqrt{(418.9)^2 + (98.8)^2}
\]
\[
\approx \sqrt{175,173.21 + 9,748.84} \approx \sqrt{184,922.05} \approx 430.55 \text{ nautical miles}
\]
**Step 7: Calculate the bearing**
To calculate the bearing from Charleston, we find the angle from the north:
\[
\text{Angle} = \tan^{-1}\left(\frac{\text{East}}{\text{North}}\right) = \tan^{-1}\left(\frac{418.9}{98.8}\right)
\]
Calculating:
\[
\approx \tan^{-1}(4.24) \approx 77.12^\circ
\]
Since this bearing is measured clockwise from north:
- The bearing is \( 21^\circ \) east of north. Therefore, the bearing from true north is \( 360 - 77.12° = 82.88°. \)
**Final Answers:**
- Bearing from Charleston, South Carolina: **82.88°**
- Distance from Charleston, South Carolina: **430.55 nautical miles**
Thus, you can round these to the nearest hundredth as follows:
Bearing: **82.88°**
Distance: **430.55 nautical miles**
1. From 1:00 a.m. to 4:30 a.m. (3.5 hours) heading due east.
2. From 4:30 a.m. to 10:30 a.m. (6 hours) heading N 21° E.
**Step 1: Calculate the distance traveled east from 1:00 a.m. to 4:30 a.m.**
The average speed of the boat is 46 knots.
The time from 1:00 a.m. to 4:30 a.m. is 3.5 hours.
\[
\text{Distance} = \text{Speed} \times \text{Time} = 46 \text{ knots} \times 3.5 \text{ hours} = 161 \text{ nautical miles}
\]
**Step 2: Calculate the boat's position at 4:30 a.m.**
At 4:30 a.m., the boat has traveled 161 nautical miles due east. We can represent this position in a coordinate system as:
- Easting: \( 161 \) nautical miles.
- Northing: \( 0 \) nautical miles.
**Step 3: Calculate the distance traveled from 4:30 a.m. to 10:30 a.m. at N 21° E.**
The time from 4:30 a.m. to 10:30 a.m. is 6 hours.
\[
\text{Distance} = \text{Speed} \times \text{Time} = 46 \text{ knots} \times 6 \text{ hours} = 276 \text{ nautical miles}
\]
**Step 4: Find the components of the distance traveled at N 21° E.**
Using trigonometry, we can break down the distance of 276 nautical miles into its eastward and northward components:
\[
\text{East Component} = 276 \times \cos(21^\circ)
\]
\[
\text{North Component} = 276 \times \sin(21^\circ)
\]
Calculating these:
- **East Component:**
\[
\text{East Component} = 276 \times \cos(21^\circ) \approx 276 \times 0.93358 \approx 257.9 \text{ nautical miles}
\]
- **North Component:**
\[
\text{North Component} = 276 \times \sin(21^\circ) \approx 276 \times 0.35837 \approx 98.8 \text{ nautical miles}
\]
**Step 5: Add the two segments together.**
Now we calculate the total east and north distances from Charleston at 10:30 a.m.:
- Total Easting:
\[
\text{Total Easting} = 161 + 257.9 \approx 418.9 \text{ nautical miles}
\]
- Total Northing:
\[
\text{Total Northing} = 0 + 98.8 \approx 98.8 \text{ nautical miles}
\]
**Step 6: Calculate the total distance from Charleston.**
Using the Pythagorean theorem to find the distance from Charleston:
\[
\text{Distance} = \sqrt{(418.9)^2 + (98.8)^2}
\]
\[
\approx \sqrt{175,173.21 + 9,748.84} \approx \sqrt{184,922.05} \approx 430.55 \text{ nautical miles}
\]
**Step 7: Calculate the bearing**
To calculate the bearing from Charleston, we find the angle from the north:
\[
\text{Angle} = \tan^{-1}\left(\frac{\text{East}}{\text{North}}\right) = \tan^{-1}\left(\frac{418.9}{98.8}\right)
\]
Calculating:
\[
\approx \tan^{-1}(4.24) \approx 77.12^\circ
\]
Since this bearing is measured clockwise from north:
- The bearing is \( 21^\circ \) east of north. Therefore, the bearing from true north is \( 360 - 77.12° = 82.88°. \)
**Final Answers:**
- Bearing from Charleston, South Carolina: **82.88°**
- Distance from Charleston, South Carolina: **430.55 nautical miles**
Thus, you can round these to the nearest hundredth as follows:
Bearing: **82.88°**
Distance: **430.55 nautical miles**