A ship sailing on a course bearing 036 degrees is 5500 metres due south of a lighthouse.If the ship continues on this course,what is the closest distance the ship will come to the lighthouse?
Well, if the ship is sailing due south of the lighthouse, it won't come any closer at all! They'll have the ultimate "social distancing" situation going on. So, the closest distance between the ship and the lighthouse will remain 5500 meters. Looks like they won't be sharing an anchor anytime soon!
To find the closest distance the ship will come to the lighthouse, you can create a right triangle with the ship, lighthouse, and the closest distance as one of the sides.
Given:
- The ship is initially 5500 meters due south of the lighthouse.
- The ship is sailing on a course bearing 036 degrees.
To solve this problem, we can assume that the ship is at the origin (0, 0) on a coordinate plane, and the lighthouse is at the point (0, -5500) on the negative y-axis.
1. Convert the bearing angle to radians:
- Bearing angle = 036 degrees
- Convert to radians: 036 * π / 180 = 0.6283 radians
2. Use trigonometry to find the x-coordinate of the closest distance:
- Use the cosine function: cosine(0.6283) = adj / hypotenuse
- The adjacent side represents the x-coordinate of the closest distance.
- The hypotenuse represents the distance between the ship and the lighthouse.
- Rearranging the equation: adj = cosine(0.6283) * hypotenuse
3. Find the hypotenuse using the distance between the ship and the lighthouse:
- The distance between the ship and the lighthouse is 5500 meters.
- The hypotenuse represents the distance between the ship and the lighthouse.
Now, plug in the values into the equation:
adj = cosine(0.6283) * 5500
4. Calculate the closest distance by taking the absolute value of the x-coordinate:
- The closest distance is the absolute value of the x-coordinate because it represents the distance.
closest_distance = |adj| = |cosine(0.6283) * 5500|
5. Calculate the closest distance:
- Substitute the value of adj into the equation:
closest_distance = |cosine(0.6283) * 5500|
Using a calculator or software to evaluate the expression, the closest distance is approximately 4561.6 meters. Therefore, the ship will come closest to the lighthouse at a distance of approximately 4561.6 meters.
To determine the closest distance the ship will come to the lighthouse, we need to find the perpendicular distance between the ship's course and the lighthouse. Here's how you can calculate it:
1. Convert the bearing to a compass direction: Since the bearing is given as 036 degrees, it represents a direction 36 degrees east of due north. In the compass system, this corresponds to a direction of 324 degrees counter-clockwise from the positive x-axis.
2. Draw a diagram: Draw a coordinate plane with the lighthouse at the origin (0, 0). Place the ship at a point (x, y) coordinates relative to the lighthouse.
3. Calculate the coordinates of the ship: Since the ship is 5500 meters due south of the lighthouse, its y-coordinate would be -5500. To find the x-coordinate, we use trigonometry.
- Decompose the motion: Since the ship is moving in a northeasterly direction at an angle of 324 degrees, we can decompose the motion into its north and east components.
- North Component: sin(324) * distance = north component
- East Component: cos(324) * distance = east component
- Calculate the x-coordinate: Since the ship's course bearing is 036 degrees (or 324 degrees clockwise), the north component and the east component will be negative. Therefore, we have:
- x-coordinate = -east component
- x-coordinate = -cos(324) * distance
4. Calculate the distance from the ship to the lighthouse: Now that we have the coordinates of the ship relative to the lighthouse, we can use the distance formula to find the shortest distance between them.
- distance = sqrt((x-coordinate)^2 + (y-coordinate)^2)
- distance = sqrt((-cos(324) * distance)^2 + (-5500)^2)
- distance = sqrt((cos(324))^2 * distance^2 + 5500^2)
5. Substitute the value of distance: Replace distance with 5500 and calculate the expression.
- distance = sqrt((cos(324))^2 * 5500^2 + 5500^2)
- distance = sqrt((cos(324))^2 * (5500^2 + 5500^2))
- distance = sqrt((cos(324))^2 * 2 * 5500^2)
- distance = 5500 * sqrt(2) * cos(324)
Therefore, the closest distance the ship will come to the lighthouse is given by the expression 5500 * sqrt(2) * cos(324) meters.
Draw a figure showing the 5500 m north-south line, the ship's path 36 degrees east of north, and the closest approach. They form a right triangle.
Closest distance = 5500 sin36
= 3233 meters