SOLUTION: A boat is sailing due east parallel to the shore line at a speed of 10 miles per hour. At a given time, the bearing of the lighthosue is 70 degrees E and 15 minutes later, the bear

Question 388063: A boat is sailing due east parallel to the shore line at a speed of 10 miles per hour. At a given time, the bearing of the lighthosue is 70 degrees E and 15 minutes later, the bearing is 63 degrees E. The lighthouse is located on the shoreline. What is the distance from the boat to the shoreline.
Found 2 solutions by CharlesG2, lwsshak3:
Answer by CharlesG2(834) (Show Source): You can put this solution on YOUR website!
A boat is sailing due east parallel to the shore line at a speed of 10 miles per hour. At a given time, the bearing of the lighthouse is 70 degrees E and 15 minutes later, the bearing is 63 degrees E. The lighthouse is located on the shoreline. What is the distance from the boat to the shoreline.

boat 10 mph
boat in 15 minutes (0.25 hours) --> d = 10 * 0.25 = 2.5 miles
bearing A = 70 degrees, bearing B = 63 degrees
difference between bearing A and bearing B = 70 - 63 = 7 degrees
boat is headed due east, bearing 90 degrees
90 degrees - 7 degrees = 83 degrees
tangent = opposite/adjacent = tangent 83 = bearing differential
opposite = distance from boat to shoreline
adjacent = 2.5 miles = reference distance
distance from shoreline to boat = 2.5 * tangent 83
20.36 rounded to 2 places miles
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
I need to draw a diagram to get started. Since the program I am using does not allow me to draw a simple diagram of the problem, I will try to describe how to construct it.
1. Draw two horizontal parallel lines about an inch apart. The top line represents the shore line and the bottom line, the direction of travel of the boat (due East)
2. Locate a point on the top horizontal line to represent the location of the lighthouse.
3. Draw a straight line from this point to the bottom line at which the point represents the position of the boat at the given time. Label the angle between this line and the top line 70 deg. From the bottom line point, draw a vertical line to the top line which completes a right triangle.
4. Draw a similar line of 63 deg which represents the position of the boat 15 minutes later.
This drawing shows we have two right triangles to work with. The first triangle has an acute angle of 70 deg with its adjacent side representing the distance the boat already traveled past the lighthouse at its given position. Call this distance x. The opposite side of the angle is the distance from boat to shoreline. Call this distance d.
The second triangle has an acute angle of 63 deg with its adjacent side representing distance the boat already traveled past the lighthouse plus the additional distance it traveled 15 minutes later. The opposite side is the same as that of the first triangle, which is the distance from boat to shoreline. Here are the calculations to find d, distance from boat to shoreline:
tan 70 = d/x (first triangle)
d = x tan 70)
after 15 min or 1/4 hour, the boat traveled (1/4)*10 = 10/4 =5/2 miles
tan 63 = d/(x+5/2) (second triangle)
d =(x+5/2) tan 63=x tan 63+(5/2) tan 63
x tan 70 = x tan 63+(5/2) tan 63
x tan 70 - x tan 63 = (5/2) tan 63
x(tan 70-tan 63) =(5/2) tan 63
x =( (5/2) tan 63)/(tan 70- tan 63)
x = 6.25 miles
d = x tan 70 =17.18 miles
ans: distance from the boat to shoreline is 17.18 miles

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