Related Rates
I am having trouble finding the equation to use to fit this all together.
At noon, ship A is 100km west of ship B. Ship A is sailing south at 35 km/s and ship B is sailing north at 25 km/s. How fast is the distance between the ships changing at 4:00pm.
just use the good old Pythagorean Theorem. If B is at (0,0) at noon, then at time t hours,
A is at (-100,-35t)
B is at (0,25t)
The distance d is thus
d^2 = 100^2 + (35t+25t)^2
= 100^2 + 3600t^2
at 4:00 pm, t=4, so
d = 260
now, we get
2d dd/dt = 7200t
dd/dt = 7200*4/(2*260) = 720/13
Or, you can consider the distance x and y, traveled by the ships A and B:
d^2 = 100^2 + (x+y)^2
2d dd/dt = 2(x+y)(dx/dt + dy/dt)
2*260 dd/dt = 2(140+100)(35+25)
dd/dt = 720/13
To solve this related rates problem, we need to find an equation that relates the changing variables. In this case, we are looking for the rate at which the distance between the ships is changing at a specific time.
Let's define the following variables:
- D: The distance between ship A and ship B at a given time (in kilometers).
- t: The time elapsed since noon (in hours).
- dD/dt: The rate at which the distance between the ships is changing (in kilometers per hour).
First, let's understand the motion of both ships.
- Ship A is sailing south at a constant speed of 35 km/hour.
- Ship B is sailing north at a constant speed of 25 km/hour.
Since we are interested in the rate of change of the distance between the ships at 4:00pm, we need to find D and t at that specific time.
At noon, the time elapsed since noon is 0 hours, and the distance between the ships is 100 km. Therefore, we have:
D = 100 km
t = 0 hours
We need to determine how the distance D changes with respect to time t. Since ship A is moving south and ship B is moving north, the distance between them will change according to their relative velocities. We can describe the change in D as:
D = D(t)
To find the equation relating D and t, we need to consider the rates of change for both ships. Since ship A is moving south and ship B is moving north, their velocities have different signs. We can express their velocities as follows:
- Velocity of ship A: -35 km/hour (negative because it is moving south).
- Velocity of ship B: +25 km/hour (positive because it is moving north).
Now, we can establish the equation relating D and t by considering the relative velocities of the ships:
D = D(t) = 100 km + [(-35 km/hour) × t + (+25 km/hour) × t]
Simplifying the equation, we have:
D = D(t) = 100 km - 10t km/hour
To find the rate at which the distance between the ships is changing (dD/dt) at 4:00pm, we differentiate the equation D(t) with respect to t:
dD/dt = d(100 km - 10t km)/dt
= -10 km/hour
Therefore, the rate at which the distance between the ships is changing at 4:00pm is -10 km/hour. The negative sign indicates that the distance is decreasing.