Two ships leave different ports A and B 100 miles apart at 0800 hours, each heading for the opposite port on reciprocal courses. Ship A steams at 20 knots and ship B at 15 knots. calculate:

(a) What time are they first 45 miles apart?
(b) What time they will pass, and how far from port A.
(c) What time they will be 30 miles apart after passing?

They approach at 35 knots. So, when they have gone 55 miles together, they are 45 miles apart. That takes 55/35 hours.

How long to consume the whole 100 miles? 100/35 hours. How far has ship A gone by then?

How long to cover 130 miles? 130/35 hours.

I assume you can convert elapsed time to time of day...

To solve this problem, we can use the formula: Distance = Speed * Time. We will also need to consider that the ships are traveling in opposite directions.

(a) To find the time when they are first 45 miles apart, we need to determine how long it will take each ship to travel a distance of 45 miles.

For Ship A:
Distance = Speed * Time
45 miles = 20 knots * Time_A
Time_A = 45 miles / 20 knots = 2.25 hours

For Ship B:
Distance = Speed * Time
45 miles = 15 knots * Time_B
Time_B = 45 miles / 15 knots = 3 hours

Since they started at the same time, we use the maximum time, which is 3 hours. Therefore, they will be first 45 miles apart after 3 hours.

(b) To find the time they will pass and how far from port A, we need to calculate when their total distances equal 100 miles. The distance traveled by Ship A is given by 20 knots * Time_A, and the distance traveled by Ship B is given by 15 knots * Time_B.

20 knots * Time_A + 15 knots * Time_B = 100 miles

Plugging in the values we calculated above:
20 knots * 2.25 hours + 15 knots * 3 hours = 45 miles + 45 miles = 90 miles

Therefore, they will pass each other after 2.25 hours, and they will be 90 miles from port A at that time.

(c) To find the time they will be 30 miles apart after passing, we need to determine how long it will take for Ship A to travel an additional 30 miles. Ship B will already be 100 - 30 = 70 miles from the port.

For Ship A, the distance from port A to the point where they are 30 miles apart is given by 20 knots * (Time_A + Time_30), where Time_30 is the additional time it takes for Ship A to travel 30 miles.

20 knots * (Time_A + Time_30) = 30 miles

We can solve this equation to find Time_30:
(Time_A + Time_30) = 30 miles / 20 knots = 1.5 hours

Therefore, they will be 30 miles apart after passing in 2.25 hours + 1.5 hours = 3.75 hours, or 3 hours and 45 minutes.