At noon, 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 30 km/h. How fast is the distance between the ships changing at 4:00 PM?

To solve this problem, we can use the concept of rates and consider the distance between the two ships as a right-angled triangle. Let's break it down step-by-step:

Step 1: Draw a diagram
Draw a diagram to visualize the scenario. Draw two ships A and B initially located 150 km apart. Ship A is moving east, and ship B is moving north.

Step 2: Define variables
Let t be the time in hours since noon.

Step 3: Calculate ship A's position
Since ship A is sailing east at a constant speed, its position can be represented as its initial position (150 km) plus its speed multiplied by the time travelled (35 km/h * t). Therefore, ship A's position at time t is 150 + 35t km east of its initial position.

Step 4: Calculate ship B's position
Since ship B is sailing north at a constant speed, its position can be represented as its initial position (0 km) plus its speed multiplied by the time travelled (30 km/h * t). Therefore, ship B's position at time t is 0 + 30t km north of its initial position.

Step 5: Calculate the distance between the ships
The distance between the ships is the hypotenuse of a right-angled triangle, with the positions of ship A and ship B being the other two sides. Using the Pythagorean theorem, the distance d(t) between the ships at time t can be calculated as:

d(t) = √((ship A's position)^2 + (ship B's position)^2)
= √((150 + 35t)^2 + (30t)^2)

Step 6: Differentiate the distance with respect to time
To find how fast the distance between the ships is changing at 4:00 PM, we need to find the derivative of the distance function d(t) with respect to time. We will differentiate using the chain rule.

d'(t) = [1 / 2√((150 + 35t)^2 + (30t)^2)] * [2(150 + 35t)(35) + 2(30t)(30)]
= [2(150 + 35t)(35) + 2(30t)(30)] / (2√((150 + 35t)^2 + (30t)^2))
= (35(150 + 35t) + 30(30t)) / √((150 + 35t)^2 + (30t)^2)

Step 7: Calculate the rate of change at 4:00 PM
To find the rate of change of the distance between the ships at 4:00 PM, we substitute t = 4 into the derivative equation.

d'(4) = (35(150 + 35 * 4) + 30(30 * 4)) / √((150 + 35 * 4)^2 + (30 * 4)^2)

Now we can calculate this value.

To find how fast the distance between the ships is changing at 4:00 PM, we can use the concept of rates of change. We need to find the rate of change of the distance between the ships with respect to time.

Let's break down the problem step by step:

1. Determine the positions of the ships at noon:
Ship A: 150 km west of Ship B
Ship B: Starting position

2. Determine the velocities of the ships:
Ship A: 35 km/h eastward
Ship B: 30 km/h northward

3. Recall that the distance between two points can be calculated using the Pythagorean theorem:
distance^2 = (change in x)^2 + (change in y)^2
In this case, the change in x represents how much Ship A travels east or west (horizontal distance), and the change in y represents how much Ship B travels north or south (vertical distance).

4. Determine the rate at which the distance changes:
We want to find the rate of change of the distance between the ships at 4:00 PM, which means we need to find how the distance is changing with respect to time. This can be calculated using the chain rule from calculus.

Here's how we can calculate it:

Step 1: Determine the change in x and y:
To find how far each ship has traveled horizontally and vertically, we need to multiply their velocities by the time (t) difference between noon and 4:00 PM, which is 4 hours.

Change in x (horizontal distance for Ship A) = 35 km/h * 4 hours = 140 km
Change in y (vertical distance for Ship B) = 30 km/h * 4 hours = 120 km

Step 2: Calculate the distance between the ships at noon:
The distance between the ships at noon is the hypotenuse of a right triangle formed by the change in x and y. By applying the Pythagorean theorem, we can find it.

distance = sqrt((change in x)^2 + (change in y)^2)
= sqrt((140 km)^2 + (120 km)^2)
= sqrt(19600 km^2 + 14400 km^2)
= sqrt(34000 km^2)
= 184.43 km

Step 3: Calculate how the distance between the ships is changing with respect to time:
We want to find the rate at which the distance is changing, so we need to differentiate the distance formula with respect to time (t).

d(distance) / d(t) = (d(distance) / d(x)) * (d(x) / d(t)) + (d(distance) / d(y)) * (d(y) / d(t))

The first term (d(distance) / d(x)) * (d(x) / d(t)) represents the change in x with respect to time, and the second term (d(distance) / d(y)) * (d(y) / d(t)) represents the change in y with respect to time.

d(distance) / d(x) = x / distance and d(distance) / d(y) = y / distance, where x is the change in x and y is the change in y.

Plugging in the values we've calculated:

d(distance) / d(t) = (x / distance) * (d(x) / d(t)) + (y / distance) * (d(y) / d(t))
= 140 km / 184.43 km * 0 km/h + 120 km / 184.43 km * 0 km/h
= 0 km/h + 0 km/h
= 0 km/h

Therefore, the distance between the ships is not changing at 4:00 PM.

the distance z between the ships after t hours, is given by

z^2 = (150-35t)^2 + (30t)^2
at t=4, z=30√5

So, plug in your values, using

2z dz/dt = -70(150-35t) + 1800t

where does the -70 come from?