Problema Solution
A sailboat and a speedboat collided recently. The speedboat was returning from a picnic on an island. The island is located 5.2 miles east and 12.6 miles north of the dock. The speedboat was heading from the island toward a point that is 3.0 miles west of the dock. At the same time the sailboat left the dock heading toward a buoy that is 5.0 miles west and 7.5 miles north of the dock. Write an equation for the path of each boat. (Show/Explain how you arrived at your equations.) Where did the boat collide? (Show/explain how you arrived at your answer.)
Answer provided by our tutors
Treat the dock as the origin on an xy-axis. North and south would be positive and negative y, respectively, and east and west would be positive and negative x.
Island location: (5.2, 12.6)
Buoy location: (-5.0, 7.5)
Speed boat headed toward (-3.0, 0) from (5.2, 12.6)
Sailboat headed toward (-5.0, 7.5) from (0, 0)
Using point-slope formula:
Sailboat equation: slope = (7.5 - 0)/(-5.0 - 0) = -1.5
y - 0 = -1.5(x - 0) => y = -1.5x
Speedboat equation: slope = (12.6 - 0)/(5.2 - -(3.0)) = 12.6/8.2 = 1.54
y - 0 = 1.54(x - -(3.0)) => y = 1.54x + 4.61
The boats collide where the equations intersect:
-1.5x = 1.54x + 4.61
3.04x = -4.61
x = -1.52
y = -1.5(-1.52) = 2.28
The boats collide 1.52 miles west and 2.28 miles north of the dock.