At noon, ship A is 40 nautical miles due west of ship B. Ship A is sailing west at 16 knots and ship B is sailing north at 17 knots. How fast (in knots) is the distance between the ships changing at 5 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)
Let the time past noon be t hours
Distance, since noon, travelled by the westbound ship is 16t nautical miles, and ship B is 17t nautical miles.
Let D be the distance between them
D^2 = (16t)^2 + (17t)^2
D^2= 545t^2
D = (√545)t
dD/dt = √545
notice that dD/dt is a constant and independent of the time
So the distance between them is constantly changing at √545 knots or 23.35 knots
To find how fast the distance between the ships is changing at 5 PM, we can use the concept of rate of change.
Let's break down the problem:
1. Ship A is 40 nautical miles due west of ship B at noon.
2. Ship A is sailing west at a speed of 16 knots.
3. Ship B is sailing north at a speed of 17 knots.
To find how fast the distance between the ships is changing at 5 PM, we need to find the rate of change of the distance between the ships with respect to time.
Let's denote:
- The distance between the ships as D (in nautical miles).
- The time elapsed since noon as t (in hours).
To find the rate of change, we need to differentiate D with respect to t.
Now, let's analyze the situation:
At noon (t = 0), the distance between the ships is 40 nautical miles, which remains constant.
After t hours, the change in the distance between the ships can be calculated by considering the velocities of both ships.
Ship A is sailing west at a constant speed of 16 knots, which implies that its position after t hours will be 16t nautical miles due west of its initial position.
Ship B is sailing north at a constant speed of 17 knots, which implies that its position after t hours will be 17t nautical miles due north of its initial position.
Using the Pythagorean theorem, we can find the distance between the two ships at a particular time t:
D^2 = (16t)^2 + (17t)^2
Simplifying this equation, we get:
D^2 = 256t^2 + 289t^2
D^2 = 545t^2
D = sqrt(545) * t
To find how fast the distance D is changing at 5 PM (t = 5), we need to differentiate D with respect to t and then substitute t = 5 into the resulting expression.
Differentiating D = sqrt(545) * t with respect to t, we get:
dD/dt = sqrt(545)
Therefore, the rate at which the distance between the ships is changing at 5 PM is sqrt(545) knots per hour.