A ship sailing on a course bearing 015 degrees is 3500 m due south of a lighthouse.
If the ship continues on this course, what is the closest distance the ship will come to the lighthouse? (Round to the nearest whole number)
if you draw a diagram, you can see that the distance you want is
3500 sin15°
To find the closest distance the ship will come to the lighthouse, we can use trigonometry. Here's how you can do it:
1. Draw a diagram: Draw a right triangle with the lighthouse at the top, the ship at the bottom, and the closest distance between them as the hypotenuse.
2. Identify the given information: The ship is 3500 meters due south of the lighthouse, and the ship's course is bearing 015 degrees.
3. Calculate the horizontal and vertical distances: Since the ship is due south of the lighthouse, the vertical distance is 3500 meters. To find the horizontal distance, we can use trigonometry. Take the sine of the angle, which is given as 015 degrees, to find the ratio between the opposite side and the hypotenuse. Multiply this ratio by the hypotenuse to find the horizontal distance. In this case, sin(015°) = 0.259. Therefore, the horizontal distance is 0.259 * 3500 = 906.5 meters.
4. Calculate the closest distance: Now that we have the horizontal and vertical distances, we can use the Pythagorean theorem to find the hypotenuse, which represents the closest distance between the ship and the lighthouse. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse squared is equal to (906.5^2 + 3500^2). Solving this equation gives us a value of 3,607,511.25. Taking the square root gives us the closest distance of approximately 1,900 meters.
Therefore, the closest distance the ship will come to the lighthouse is approximately 1,900 meters.
To find the closest distance the ship will come to the lighthouse, we need to determine the perpendicular distance between the ship's course and the lighthouse.
First, let's convert the ship's course bearing from degrees to radians.
15 degrees is equal to (15 * π) / 180 radians.
Now, let's take the sine of this angle to find the perpendicular distance.
Perpendicular distance = 3500 m * sin((15 * π) / 180)
≈ 900.96 m
Rounding to the nearest whole number, the closest distance the ship will come to the lighthouse is approximately 901 meters.