When navigating their crafts, ship captains and airplane pilots can often be seen drawing lines on a large map?
A cruise ship is traveling in the Atlantic Ocean at a constant rate of
40 mi/h and is traveling 2 mi east for every 5 mi north. An oil tanker is
350 mi due north of the cruise ship and is traveling 1 mi east for every 1
mi south.
a. How far is each ship from the point at which their paths cross?
b. What rate of speed for the oil tanker would put it on a collision
course with the cruise ship?
If you look at the response to your previous question on airplanes, you may have a better idea how this similar question can be tackled.
http://www.jiskha.com/display.cgi?id=125415001
To answer these questions, we need to solve two related problems: finding the point at which the ships' paths cross and determining the distance between each ship and that point. Let's go step by step.
Step 1: Find the point at which their paths cross.
To do this, we need to represent the paths of the two ships as equations. Let's call the initial position of the cruise ship (0,0).
For the cruise ship, we know it travels 2 miles east for every 5 miles north. So, the equation for the cruise ship's path is y = (5/2)x, where x represents the distance traveled east and y represents the distance traveled north.
For the oil tanker, we know it travels 1 mile east for every 1 mile south. So, the equation for the oil tanker's path is x = -y, where x represents the distance traveled east and y represents the distance traveled north.
To find the point of intersection, we need to solve the system of equations:
(5/2)x = y (equation for the cruise ship's path)
x = -y (equation for the oil tanker's path)
We can solve this system of equations by substituting the value of x from the second equation into the first equation:
(5/2)(-y) = y
-5y/2 = y
-5y = 2y
7y = 0
y = 0
Substituting the value of y back into the second equation:
x = 0
So, the point at which their paths cross is (0,0).
Step 2: Determine the distance between each ship and the point of intersection.
To find the distance, we can use the distance formula:
Distance = √((x2 - x1)² + (y2 - y1)²)
For the cruise ship, the coordinates of its position with respect to the point of intersection are (0, 0). So, the distance between the cruise ship and the point of intersection is:
Distance (cruise ship) = √((0 - 0)² + (0 - 0)²) = √(0 + 0) = 0
For the oil tanker, the coordinates of its position with respect to the point of intersection are (0 - 350, 0). So, the distance between the oil tanker and the point of intersection is:
Distance (oil tanker) = √((-350 - 0)² + (0 - 0)²) = √((-350)²) = √(122,500) = 350
a. How far is each ship from the point at which their paths cross?
The distance between the cruise ship and the point of intersection is 0 miles, and the distance between the oil tanker and the point of intersection is 350 miles.
b. What rate of speed for the oil tanker would put it on a collision course with the cruise ship?
Since the distance between the cruise ship and the point of intersection is 0, the oil tanker is already on a collision course with the cruise ship. Thus, any rate of speed will eventually lead to a collision.