A navy destroyer is exactly 68 miles North of an aircraft carrier, which is travelling West 21 degrees North at 18 mph. what speed and bearing must the destroyer maintain if it is to reach the aircraft carrier in exactly 3 hrs?
and what is your thinking? I will be happy to critique it.
would it be 226 mph
???/
To determine the speed and bearing that the destroyer must maintain in order to reach the aircraft carrier, we can break down the problem into two components: the northward component and the westward component.
1. Northward Component:
- The destroyer is exactly 68 miles north of the aircraft carrier.
- The time to reach the aircraft carrier is given as 3 hours.
- Therefore, the required speed of the destroyer along the northward direction can be calculated as:
Speed = Distance / Time = 68 miles / 3 hours = 22.67 mph (approximately).
2. Westward Component:
- The aircraft carrier is traveling west at 18 mph, which creates a westward component.
- To cancel out the westward component, the destroyer needs to have an equal and opposite eastward component.
- This can be achieved by applying trigonometry using the given angle (21 degrees north) and the northward speed we calculated earlier.
- Let's find the eastward component:
Eastward Component = Northward Component x tan(angle)
Eastward Component = 22.67 mph x tan(21 degrees) = 8.11 mph (approximately).
Now let's combine the two components to find the resultant speed and bearing:
- Resultant Speed: The resultant speed is the vector sum of the northward and eastward components. We can find it using the Pythagorean theorem:
Resultant Speed = √(Northward Component^2 + Eastward Component^2)
Resultant Speed = √(22.67^2 + 8.11^2) = √(515.17 + 65.92) = √581.09 = 24.10 mph (approximately).
- Resultant Bearing: The resultant bearing is the direction in which the destroyer must travel. We can find it using trigonometry:
Resultant Bearing = arctan(Eastward Component / Northward Component)
Resultant Bearing = arctan(8.11 mph / 22.67 mph) = arctan(0.357) = 19.95 degrees (approximately).
Therefore, the destroyer must maintain a speed of approximately 24.10 mph and a bearing of approximately 19.95 degrees (northward of due west) in order to reach the aircraft carrier in exactly 3 hours.