One ship is 20 miles due North of another ship, and is sailing South at the rate of 10 miles per hour. The second ship sails West at the rate of 20 miles per hour. For how long will the ships continue to approach each other?
To find out how long the ships will continue to approach each other, we need to determine when they will intersect.
Let's break down the problem step by step:
1. Draw a diagram: Draw two ships labeled Ship A and Ship B. Ship A is 20 miles north of Ship B.
2. Assign variables: Let t represent the time in hours for which the ships will continue to approach each other.
3. Determine Ship A's distance: Since Ship A is sailing South at a speed of 10 miles per hour for t hours, its distance traveled can be calculated as 10t miles.
4. Determine Ship B's distance: Since Ship B is sailing West at a speed of 20 miles per hour for t hours, its distance traveled can be calculated as 20t miles.
5. Calculate the distance between the ships: At any given time t, the distance between the ships can be calculated using the Pythagorean theorem as follows:
- Ship A's distance traveled = 10t miles (South)
- Ship B's distance traveled = 20t miles (West)
- According to the Pythagorean theorem, the distance between the ships is given by √[(10t)^2 + (20t)^2].
6. Set up the equation: Since the ships are approaching each other, their distance is decreasing. Therefore, we need to find a time t when the distance between them is 0. So, we set up the equation:
√[(10t)^2 + (20t)^2] = 0
7. Solve the equation: Square both sides of the equation to remove the square root:
[(10t)^2 + (20t)^2] = 0
Simplify the equation by expanding and combining like terms:
100t^2 + 400t^2 = 0
Combine like terms:
500t^2 = 0
Divide both sides by 500:
t^2 = 0
Take the square root of both sides:
t = √0
The time is t = 0.
Since the time t is 0, it means that the ships will continue to approach each other until they intersect, which is at the starting point. In other words, they will continue to approach each other indefinitely.