Two cruise ships leave the same point outside St. John’s harbor at noon. Ship A travels west at 20 km/hr, while Ship B travels south at 25 km/hr. Using calculus, determine how fast they are separating from each other at 2:00 p.m.

draw a diagram. At time t hours, the distance z is

z^2 = (20t)^2 + (25t)^2

now just figure dz/dt at t=2.

To determine how fast the two cruise ships are separating at 2:00 p.m., we need to find the rate of change of the distance between them with respect to time.

Let's denote the position of Ship A as (x, 0) and the position of Ship B as (0, y), where x and y are the distances traveled by the ships in hours.

The distance between the ships at any given time can be determined using the distance formula:

Distance^2 = (x - 0)^2 + (0 - y)^2

Distance = √(x^2 + y^2)

To find the rate of change of the distance between the ships, we need to differentiate this expression with respect to time (t):

d(Distance)/dt = d(√(x^2 + y^2))/dt

To evaluate this derivative, we can apply the chain rule. We need to find the derivative of √(x^2 + y^2) with respect to x and y, and then multiply by the rate of change of x with respect to t and the rate of change of y with respect to t.

First, we differentiate √(x^2 + y^2) with respect to x:

d(√(x^2 + y^2))/dx = (1/2)(x^2 + y^2)^(-1/2) * 2x
= x/sqrt(x^2 + y^2)

Next, we differentiate √(x^2 + y^2) with respect to y:

d(√(x^2 + y^2))/dy = (1/2)(x^2 + y^2)^(-1/2) * 2y
= y/sqrt(x^2 + y^2)

Finally, we multiply these derivatives by the rates of change of x and y with respect to t:

d(Distance)/dt = (x/sqrt(x^2 + y^2)) * dx/dt + (y/sqrt(x^2 + y^2)) * dy/dt

Now we can substitute the given rates of change:

dx/dt = 20 km/hr (rate of change of x with respect to t)
dy/dt = -25 km/hr (rate of change of y with respect to t, with a negative sign indicating southward direction)

Plugging in these values:

d(Distance)/dt = (x/sqrt(x^2 + y^2)) * 20 + (y/sqrt(x^2 + y^2)) * (-25)

To find the specific rate at 2:00 p.m., we can substitute x = 2 and y = 2:

d(Distance)/dt = (2/sqrt(2^2 + 2^2)) * 20 + (2/sqrt(2^2 + 2^2)) * (-25)

Simplifying:

d(Distance)/dt = (2/√8) * 20 - (2/√8) * 25

d(Distance)/dt = (10√2 - 25√2) km/hr

d(Distance)/dt = (10 - 25)√2 km/hr

d(Distance)/dt = -15√2 km/hr

Therefore, the two cruise ships are separating from each other at a rate of 15√2 km/hr at 2:00 p.m.