Answer:
To find how fast the two ships are separating at 5:45 p.m., we can use the concept of related rates. Let's break down the problem step by step:
1. The first ship is sailing to the east at 10 mph. So, its horizontal (eastward) speed is 10 mph, and its vertical (southward) speed is 0 mph.
2. The second ship is sailing due south at 15 mph. Its horizontal speed is 0 mph, and its vertical (southward) speed is 15 mph.
3. We need to find the separation rate between these two ships at 5:45 p.m. To do this, we'll calculate the rates of change of their positions at that specific time.
4. Let's use the coordinates to represent the positions of the ships at any time. We'll set up a coordinate system where the lighthouse is at the origin (0,0). The position of the first ship at any time can be represented as (10t, 0), where t is the time in hours since 3:15 p.m., and the position of the second ship can be represented as (0, 15(t - 1/2)), where t is the time in hours since 3:45 p.m.
5. To find the separation between the two ships at 5:45 p.m. (t = 2.5), we can use the distance formula:
Separation, D = √((10t - 0)^2 + (0 - 15(t - 1/2))^2)
6. Now, let's calculate the separation rate, dD/dt, by differentiating the above expression with respect to time t:
dD/dt = 1/2 * (2 * (10t - 0) * 10 + 2 * (0 - 15(t - 1/2)) * (-15))
7. Simplify the expression:
dD/dt = 100t - 225(t - 1/2)
8. Substitute t = 2.5 into the equation to find the separation rate at 5:45 p.m.:
dD/dt = 100 * 2.5 - 225 * (2.5 - 1/2)
9. Calculate the value:
dD/dt = 250 - 225 * 2
10. Now, compute the final answer, rounding it to three decimal places:
dD/dt ≈ 250 - 450 = -200
So, at 5:45 p.m., the two ships are separating at a rate of 200 mph, and since the ships are moving away from each other, the negative sign indicates that the separation is in the south and east directions.