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?
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To answer these questions, we need to draw a diagram on a map. Let's follow these steps to solve the problem:
1. Start by drawing a large-scale map that represents the Atlantic Ocean. Make sure to include a scale so that we can measure distances accurately.
2. Mark the starting point of the cruise ship and label it as "C". Also, mark the starting point of the oil tanker and label it as "O".
3. Using the given information, draw the path of the cruise ship on the map. Since it travels 2 miles east for every 5 miles north, you need to draw a line that slopes upward and to the right. Each segment of the line should be 5 units long vertically and 2 units long horizontally. Label the endpoint of this line as "P" for the point of intersection.
4. Draw the path of the oil tanker on the map. Since it travels 1 mile east for every 1 mile south, you need to draw a line that slopes downward and to the right. Each segment of the line should be 1 unit long vertically and 1 unit long horizontally. Label the endpoint of this line as "P".
5. Now, we have two triangles: triangle COP (cruise ship, oil tanker, point of intersection) and triangle CPO.
a. To find the distance between each ship and the point of intersection, we need to calculate the length of the sides of triangle COP. You can use the Pythagorean theorem to do this. The cruise ship's distance is 350 miles north and 2 miles east, which gives us a right-angled triangle. Using the theorem, you can calculate the length of the hypotenuse.
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 the slope of line "OP" (the path of the oil tanker) between points "O" and "P". This will give us the rate at which the oil tanker is moving southeast. From there, we can calculate the speed needed to reach the point of intersection.
Remember to convert units if necessary and be mindful of directions and angles when working with maps.