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 woul put it on a collision course with the cruise ship?
For the path-crossing point, you don't need velocities. Let Y be miles north and X = miles east. Let (0,0) be the coordinates of the cruise ship initially.
Y = 350 - X (for tanker)
Y = (5/2) X (for cruise ship)
The cross when
0 = 350 - (7/2)X
X = 100 miles
Y = 250 miles
a) The cruise ship is sqrt [(100)^2 + (250)^2] = 269.3 miles from that point initially. The tanker is
sqrt [(100)^2 + (100)^2] = 141.4 miles away.
b) First calculate when the cruise ship arrives at the crossing point, based upon its known speed:
T = 269.3/40 = 6.73 hours
Then calculate the speed that would put the tanker at that point at that time
V = 141.4 miles/T
B is about 21 mi/h
T = 269.3/40 = 6.73 hours
the you have to 141.4/6.73 which equals 21 mi/h
To find the distance between the ships and the point of intersection, we can use the Pythagorean theorem:
a) Distance between cruise ship and point of intersection:
Let x be the distance the cruise ship travels north.
The distance the cruise ship travels east would then be (2/5) * x.
We can use the Pythagorean theorem to find the distance between the cruise ship and the point of intersection:
Distance = √((2/5 * x)^2 + x^2)
Distance between oil tanker and point of intersection:
The oil tanker is 350 miles directly north of the cruise ship, so the distance between the oil tanker and the point of intersection can be calculated as:
Distance = √(x^2 + (x - 350)^2)
b) To find the rate of speed for the oil tanker that would put it on a collision course with the cruise ship, we need to determine when the distances between the ships and the point of intersection are equal.
So, we equate the two distance formulas:
√((2/5 * x)^2 + x^2) = √(x^2 + (x - 350)^2)
Squaring both sides:
((2/5 * x)^2 + x^2) = (x^2 + (x - 350)^2)
Simplifying:
(4/25 * x^2 + x^2) = (x^2 + (x - 350)^2)
Combining like terms:
(29/25 * x^2) = (x^2 + (x - 350)^2)
Rearranging the equation:
0 = (x^2 + (x - 350)^2) - (29/25 * x^2)
Now, we can solve this equation to find the value of x that represents the distance from the point of intersection.
To solve this problem, we can break it down into two parts: finding the point where the two paths intersect and then calculating the distances and speeds.
a. Finding the Point of Intersection:
Let's consider the coordinates of the two ships at a given time. Assume that they meet after time t.
For the cruise ship:
Distance traveled north = Speed * Time = (40 mi/h) * t
Distance traveled east = 2 mi for every 5 mi north = (2/5) * (40 mi/h) * t
For the oil tanker:
Distance traveled north = 350 mi + Distance traveled south = 350 mi + (1 mi/south) * t
Distance traveled east = 1 mi for every 1 mi south = (1/1) * (1 mi/south) * t
To find the point of intersection, we equate the distances traveled north and east for both ships:
(40 mi/h) * t * (2/5) = (350 mi + t) * (1/1)
Simplifying the equation, we get:
8t = 350 + t
7t = 350
t = 350/7
t = 50
Therefore, the two ships will intersect after 50 hours.
b. Calculating Distances and Speeds:
We can now substitute t = 50 back into the equations for each ship to determine the distances from the point of intersection.
For the cruise ship:
Distance traveled north = (40 mi/h) * (50 h) = 2000 miles
Distance traveled east = (2/5) * (40 mi/h) * (50 h) = 400 miles
For the oil tanker:
Distance traveled north = 350 mi + (1 mi/south) * (50 h) = 350 mi - 50 mi = 300 miles
Distance traveled east = (1/1) * (1 mi/south) * (50 h) = 50 miles
To find the distance between the two ships, we can use the Pythagorean theorem:
Distance = √[(Distance north)^2 + (Distance east)^2]
For the cruise ship:
Distance = √[(2000^2) + (400^2)] ≈ 2032.3 miles
For the oil tanker:
Distance = √[(300^2) + (50^2)] ≈ 303.4 miles
Finally, to calculate the speed of the oil tanker required for a collision course, we divide the distance between the two ships by the time it takes for them to meet:
Required speed = (Distance between ships) / (Time taken to meet)
Required speed = (2032.3 miles + 303.4 miles) / (50 h)
Required speed ≈ 47.1 mi/h
Thus, a speed of approximately 47.1 miles per hour for the oil tanker would put it on a collision course with the cruise ship.