At 3 P.M, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 7 P.M.? (Round your answer to one decimal place.)

To solve this problem, we can use the concept of related rates. We want to find how fast the distance between the ships is changing, so we need to find the rate of change of the distance with respect to time.

Let's break down the problem and start by finding the position of each ship at 3 P.M.:

- Ship A is 150 km west of Ship B.
- Ship A is sailing east at 35 km/h, which means it is moving towards Ship B.
- Ship B is sailing north at 25 km/h, which means it is moving away from Ship A.

Now, let's define some variables to help us solve the problem:
- Let x be the distance (in km) between Ship A and Ship B at 3 P.M.
- Let y be the distance (in km) Ship A has traveled east from its initial position at 3 P.M.
- Let z be the distance (in km) Ship B has traveled north from its initial position at 3 P.M.
- Let t be the time (in hours) that has elapsed since 3 P.M.

From the given information, we know that:
- x = 150 km (since Ship A is 150 km west of Ship B at 3 P.M.)
- y = 35 km/h * t (since Ship A is traveling east at a constant speed of 35 km/h)
- z = 25 km/h * t (since Ship B is traveling north at a constant speed of 25 km/h)

Now, let's find the expression for the distance between the two ships (x) in terms of y and z:

Using the Pythagorean theorem, we have:
x^2 = y^2 + z^2

Differentiating both sides of the equation with respect to time t, we get:
2x(dx/dt) = 2y(dy/dt) + 2z(dz/dt)

Rearranging the equation, we have:
dx/dt = (y(dy/dt) + z(dz/dt)) / x

Substituting the values of y, z, and x, we get:
dx/dt = (35 km/h * t * (dy/dt) + 25 km/h * t * (dz/dt)) / 150 km

The problem asks us to find how fast the distance between the ships is changing at 7 P.M., which is 4 hours after 3 P.M. (t = 4). We need to find dy/dt and dz/dt at t = 4.

- For Ship A, dy/dt = 35 km/h (since it is moving at a constant speed of 35 km/h)
- For Ship B, dz/dt = 25 km/h (since it is moving at a constant speed of 25 km/h)

Substituting the values into the equation, we have:
dx/dt = (35 km/h * 4 hours * 35 km/h + 25 km/h * 4 hours * 25 km/h) / 150 km

Calculating the expression, we get:
dx/dt = (140 km * 35 km/h + 100 km * 25 km/h) / 150 km

Simplifying the expression, we have:
dx/dt = (4900 km^2/h + 2500 km^2/h) / 150 km

Adding the terms, we get:
dx/dt = 7400 km^2/h / 150 km

Simplifying further, we have:
dx/dt = 49.3 km/h

Therefore, the distance between the ships is changing at a rate of 49.3 km/h at 7 P.M.