At noon, ship A is 150 km west of ship B. Ship A is sailing east at 25 km/h and ship B is sailing north at 20 km/h. How fast is the distance between the ships changing at 4:00 PM?
If we assume A is at (0,0) at noon, the the distance z is
z^2 = (150-25t)^2 + (20t)^2
now get z when t=4
and find dz/dt from
2z dz/dt = -50(150-25t) + 40(20t)
To find how fast the distance between the ships is changing at 4:00 PM, we can use the concept of relative motion.
Let's first calculate the positions of the ships at 4:00 PM, considering that they started at noon.
The time difference between noon and 4:00 PM is 4 hours. Ship A has been moving for 4 hours at a constant speed of 25 km/h, so it has covered a distance of 4 hours * 25 km/h = 100 km to the east from its initial position. Therefore, the position of ship A at 4:00 PM would be 100 km to the east from its starting position.
Similarly, ship B has been moving for 4 hours at a constant speed of 20 km/h, so it has covered a distance of 4 hours * 20 km/h = 80 km to the north from its initial position. Therefore, the position of ship B at 4:00 PM would be 80 km to the north from its starting position.
Now, we can draw a right-angled triangle using the positions of ship A and ship B at 4:00 PM. The distance between the ships is the hypotenuse of this triangle, and we need to find its rate of change.
Since ship A and ship B are forming a right angle triangle, we can use the Pythagorean theorem to find the distance between them:
Distance^2 = (Distance in the east-west direction)^2 + (Distance in the north-south direction)^2
Let's assign the east-west distance as x and the north-south distance as y. So, Distance^2 = x^2 + y^2
At 4:00 PM, ship A has moved 100 km eastward, so the east-west distance becomes x = 150 km + 100 km = 250 km.
Ship B has moved 80 km northward, so the north-south distance becomes y = 80 km.
Now, we can substitute these values into the equation Distance^2 = x^2 + y^2:
Distance^2 = 250^2 + 80^2
Simplifying this equation gives us: Distance^2 = 62500 + 6400 = 68900
Now, taking the square root of both sides: Distance = √68900 ≈ 262.42 km
Now, to find how fast the distance is changing, we need to calculate the derivative of the distance equation with respect to time.
Differentiating both sides of the equation Distance^2 = 250^2 + 80^2 with respect to time gives us:
2 * Distance * (d(Distance)/dt) = 2 * 250 * (dx/dt) + 2 * 80 * (dy/dt)
Since we are interested in finding d(Distance)/dt, we can rearrange this equation:
(d(Distance)/dt) = (250 * (dx/dt) + 80 * (dy/dt)) / Distance
The value of dx/dt can be given as 25 km/h (since Ship A is moving east at 25 km/h)
and the value of dy/dt can be given as 20 km/h (since Ship B is moving north at 20 km/h).
Plugging these values into the equation, we get:
(d(Distance)/dt) = (250 * (25 km/h) + 80 * (20 km/h)) / (262.42 km)
Calculating this expression, we find:
(d(Distance)/dt) ≈ (6250 km/h + 1600 km/h) / (262.42 km)
(d(Distance)/dt) ≈ 7850 km/h / 262.42 km
(d(Distance)/dt) ≈ 29.91 km/h
Therefore, the rate at which the distance between the ships is changing at 4:00 PM is approximately 29.91 km/h.