"At 9am ship A is 50 km east of ship B. Ship A is sailing north at 40km/h and ship B is sailing south at 30km/h. How fast is the distance between them changing at noon?"
I never know how to set questions like this up!
It probably helps to sketch the ships.
The distance between the ships is a right triangle with one leg (the east/west leg) being a constant 50km and the other leg being the TOTAL distance traveled by both ships. Express the distance as a function of time then take the derivative with respect to time to find the rate of change of the distance at a given time.
To set up this problem, we need to consider the rates at which the positions of the two ships are changing.
Let's define some variables:
- Let x be the distance, in km, between ship A and ship B.
- Let t be the time in hours.
From the problem statement, we know the following:
- At 9 am (t=0), ship A is 50 km east of ship B. So initially, x(0) = 50 km.
- Ship A is sailing north at a speed of 40 km/h, so the rate of change of the position of ship A, dx/dt, is 40 km/h.
- Ship B is sailing south at a speed of 30 km/h, so the rate of change of the position of ship B, dy/dt, is -30 km/h (negative because it is in the opposite direction).
We want to find how fast the distance between the two ships is changing at noon (t=3 hours), which is given by d(x - y)/dt.
To find this, we need to first express the distance x in terms of t, and then differentiate it with respect to time.
Since ship A is moving in the north direction and ship B in the south direction, the distance between them is given by x - y.
From the information given, we can set up the equation:
x(t) = 50 + (40t) (since ship A is initially 50 km east of ship B and is moving north at a rate of 40 km/h)
Now, differentiate both sides of the equation with respect to t:
x'(t) = 0 + 40
So, dx/dt = 40 km/h.
Now we can find how fast the distance between them is changing at noon (t=3 hours) using:
d(x - y)/dt = dx/dt - dy/dt
d(x - y)/dt = 40 km/h - (-30 km/h)
d(x - y)/dt = 40 km/h + 30 km/h
d(x - y)/dt = 70 km/h
Therefore, the distance between the two ships is changing at a rate of 70 km/h at noon.