A ship is sailing due north at 12km/h while another ship is observed 15km ahead, traveling due east at 9km/h. What is the closest distance of approach of the two ships?
at time t hours after the observation, the distance z between the ships is
z^2 = (15-12t)^2 + (9t)^2
2z dz/dt = 90(5t-4)
dz/dt=0 when t = 4/5
z(4/5) = 81
Well, it seems like the two ships are playing a game of "Who can come the closest?"! Let's see if we can help them out.
Since one ship is sailing due north and the other due east, they are on a collision course. The ship traveling north is moving at 12 km/h, and the ship traveling east is moving at 9 km/h.
To find the closest distance of approach, we can use some mathematical trickery. We have a right triangle formed by the two ships and the closest distance between them. The hypotenuse of this triangle is the distance between the two ships, which is 15 km. The sides of the triangle are the speeds of the ships. So, we can use the Pythagorean theorem (a^2 + b^2 = c^2) to find the closest distance.
Let's call the closest distance between the ships "d". We know that one of the sides of the triangle is 12 km/h, and the other side is 9 km/h. Using some math magic, we have:
12^2 + 9^2 = d^2
144 + 81 = d^2
225 = d^2
Taking the square root of both sides, we find:
d = 15
Tada! The closest distance of approach between these ships is 15 km. Looks like they're getting pretty cozy out there on the open ocean!
To find the closest distance of approach between two ships, we can use the concept of relative velocity.
Step 1: Draw a diagram to visualize the situation. Let's label the northbound ship as Ship A and the eastbound ship as Ship B.
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Ship B
-------------------> (East)
|
|
|
Ship A (North)
```
Step 2: Determine the north and east velocities of each ship.
- Ship A is moving due north at a speed of 12 km/h.
- Ship B is moving due east at a speed of 9 km/h.
Step 3: Calculate the relative velocity between the ships.
We need to find the vector sum of the velocities of Ship A and Ship B.
The relative velocity between the two ships can be calculated using the Pythagorean theorem:
Relative velocity (Vrel) = √((Velocity of Ship A)^2 + (Velocity of Ship B)^2)
Vrel = √((12 km/h)^2 + (9 km/h)^2)
= √(144 km^2/h^2 + 81 km^2/h^2)
= √(225 km^2/h^2)
= 15 km/h
Step 4: Determine the time of closest approach.
To find the time it takes for the ships to be at their closest distance, we need to divide the distance between the ships by the relative velocity.
Distance between the ships = 15 km (given)
Relative velocity = 15 km/h (calculated in Step 3)
Time of closest approach = Distance / Relative velocity
= 15 km / 15 km/h
= 1 hour
Step 5: Calculate the closest distance of approach.
Since Ship A is moving north and Ship B is moving east, after 1 hour, Ship A will have traveled 12 km north and Ship B will have traveled 9 km east.
The closest distance of approach between the two ships can be found using the Pythagorean theorem:
Closest distance of approach = √((Distance traveled by Ship A)^2 + (Distance traveled by Ship B)^2)
Closest distance of approach = √((12 km)^2 + (9 km)^2)
= √(144 km^2 + 81 km^2)
= √(225 km^2)
= 15 km
Therefore, the closest distance of approach of the two ships is 15 km.
To find the closest distance of approach between the two ships, we can use the concept of relative velocity. In this case, we can consider the horizontal component of motion of the ship traveling north and the vertical component of motion of the ship traveling east.
Let's break down the problem step by step:
1. Determine the relative velocity of the ships:
- The ship traveling north has a velocity of 12 km/h in the north direction.
- The ship traveling east has a velocity of 9 km/h in the east direction.
- To find the relative velocity, we can use vector addition. Since the ships are moving at right angles to each other, we can use the Pythagorean theorem to find the magnitude of the relative velocity:
Relative velocity = √((12 km/h)^2 + (9 km/h)^2)
2. Determine the time it takes for the ships to reach the closest distance of approach:
- We know that distance = speed × time. Since the ship traveling north is initially 15 km ahead, it needs to travel a distance of 15 km to reach the closest distance of approach.
- To find the time it takes for the ship traveling north to cover this distance, we can use the formula: time = distance / speed. Therefore, time = 15 km / 12 km/h.
3. Calculate the distance of approach:
- The distance of approach can be calculated using the formula: distance = speed × time.
- Inserting the values into the formula: distance = 9 km/h × (15 km / 12 km/h).
4. Simplify the expression and calculate the result:
- Canceling out km/h in the expression will leave us with distance in kilometers.
- Simplifying the expression, we get: distance = 9 km/h × (15/12).
- Calculating this expression, we find that the closest distance of approach is 11.25 kilometers.
Therefore, the closest distance of approach between the two ships is 11.25 kilometers.