A man sails with his boat starting (x) kilometer away from the main station due north. After 3 km of sailing, he follows an arc path. When he is somewhere in between North and the West Point at P(4,yp), the engine suddenly stops. With the help of his navigational tool, he realized that he is 13.86 kilometer away from the point where he started following the arc path. How far is he from the point he started following the Arc path? starting point ?main station?
I have no idea what you mean by
somewhere in between North and the West Point
and is "yp" supposed to mean something like yp?
I dont know either. The problem was given by our teacher. And yes, it is supposes to be y subscript p. Thank you.
*supposed I mean
To solve this problem, we can break it down into two steps. First, let's find the distance between the starting point and the point where the boat started following the arc path. Then, we'll find the distance between the main station and the starting point.
Step 1: Finding the distance between the starting point and the point where the boat started following the arc path.
Given that the boat sailed 3 km due north before following the arc path, we can treat this part of the journey as a straight-line segment. This means that the boat traveled in a straight line from its original starting point (x) to point A, where it started following the arc path.
The distance traveled in the straight-line segment can be found using the Pythagorean theorem:
Distance = √((Change in x)^2 + (Change in y)^2)
In this case, the change in x is 0 km (as the boat traveled north) and the change in y is 3 km. Therefore:
Distance = √((0 km)^2 + (3 km)^2) = √(0 + 9) = √9 = 3 km
So, the distance between the starting point and the point where the boat started following the arc path is 3 km.
Step 2: Finding the distance between the main station and the starting point.
Now, to find the distance between the main station and the starting point, we can use the given information that the boat is 13.86 km away from its starting point following the arc path (P(4, yp)). Here, we have the x-coordinate of the boat's current position (4 km), but the y-coordinate (yp) is unknown.
We can use the distance formula to solve for the y-coordinate:
Distance = √((Change in x)^2 + (Change in y)^2)
Given that the change in x is 4 km and the overall distance is 13.86 km:
13.86 km = √((4 km)^2 + (Change in y)^2)
Squaring both sides of the equation, we get:
(13.86 km)^2 = (4 km)^2 + (Change in y)^2
152.7396 km^2 = 16 km^2 + (Change in y)^2
136.7396 km^2 = (Change in y)^2
Taking the square root of both sides, we find:
√(136.7396 km^2) = |Change in y|
Therefore, Change in y = ±√(136.7396 km^2)
Now, to determine the y-coordinate of the boat's current position, we need to consider the direction of the arc path. Since the boat is between North and the West Point (P(4, yp)), it means the boat is moving clockwise.
In the clockwise direction, the y-coordinate should be negative. Therefore, the y-coordinate is:
yp = -√(136.7396 km^2)
So, the distance between the main station and the starting point is the hypotenuse formed by the x and y coordinates:
Distance = √((Change in x)^2 + (Change in y)^2)
Distance = √((4 km)^2 + (-√(136.7396 km^2))^2)
Calculating this, we find:
Distance ≈ 8.57 km
Therefore, the boat is approximately 8.57 km away from the point where it started following the arc path, and the starting point itself is 3 km away from the main station.