At noon, ship A is 20 nautical miles due west of ship B. Ship A is sailing west at 21 knots and ship B is sailing north at 18 knots. How fast (in knots) is the distance between the ships changing at 5 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

at t=5

A west of origin 20+21*5 = 125 or x=-125
B north of origin = 18*5 = 90 or y = 90
h^2 = x^2+y^2
h=(x^2+y^2)^.5 = 154 where h is hypotenuse

we want dh/dt
2 h dh/dt = 2 x dx/dt + 2 y dy/dt

2(154) dh/dt = 2(-125)(-21) + 2 (90)(18)

Ahoy there! Well, it seems like we have a nautical math problem on our hands. Let's dive in and see what we can come up with, shall we?

At noon, ship A is 20 nautical miles due west of ship B. Ship A is sailing west at 21 knots, while ship B is sailing north at 18 knots.

Now, at 5 PM, we need to find how fast the distance between the two ships is changing. To do that, let's break it down step by step.

First, let's find the position of each ship at 5 PM. From noon to 5 PM, a total of 5 hours have passed.

Ship A has been sailing west at a constant speed of 21 knots for 5 hours, so it has traveled 21 knots/hour x 5 hours = 105 nautical miles west.

Ship B, on the other hand, has been sailing north at a constant speed of 18 knots for 5 hours, so it has traveled 18 knots/hour x 5 hours = 90 nautical miles north.

Now, let's find the distance between the two ships at 5 PM. Since Ship A was originally 20 nautical miles due west of Ship B, and Ship A has traveled 105 nautical miles west, we can use the good ol' Pythagorean theorem to find the distance between the two ships:

Distance^2 = (105 nautical miles)^2 + (20 nautical miles + 90 nautical miles)^2

Distance^2 = 11025 nautical miles^2 + 110 nautical miles^2

Distance^2 = 11135 nautical miles^2

Taking square roots on both sides, we find that the distance between the two ships at 5 PM is approximately 105.60 nautical miles.

Finally, we need to find how fast this distance is changing. So we differentiate the equation we used earlier with respect to time:

2 * Distance * (d(Distance)/dt) = 2 * 105.60 nautical miles * (d(Distance)/dt)

Since the rate at which the distance between the two ships is changing is what we're looking for, let's solve for (d(Distance)/dt):

(d(Distance)/dt) = (Distance * (d(Distance)/dt)) / Distance

(d(Distance)/dt) = (105.60 nautical miles * (d(Distance)/dt)) / 105.60 nautical miles

(d(Distance)/dt) = (d(Distance)/dt)

*tada!* The rate at which the distance between the two ships is changing at 5 PM is the same as the rate at which the distance between the two ships is changing at 5 PM. Am I right, or am I right?

In simpler terms (without all the math mumbo-jumbo), the speed at which the distance between the two ships is changing at 5 PM is the same as the original rate of change. Therefore, it is 105.60 knots.

Hope this clears things up for you! Fair winds and following seas!

To find the speed at which the distance between the ships is changing at 5 PM, we can use the concept of rates of change.

Let's first determine the positions of ship A and ship B at 5 PM.

From noon to 5 PM, there are 5 hours. Ship A has been sailing west at 21 knots for 5 hours, so it has traveled a distance of 21 knots/hour * 5 hours = 105 nautical miles due west.

Similarly, ship B has been sailing north at 18 knots for 5 hours, so it has traveled a distance of 18 knots/hour * 5 hours = 90 nautical miles due north.

At 5 PM, the distance between the ships is the hypotenuse of a right-angled triangle formed by the positions of A and B. This distance can be found using the Pythagorean theorem: distance^2 = (105 nautical miles)^2 + (90 nautical miles)^2.

distance^2 = 11025 + 8100
distance^2 = 19125
distance = sqrt(19125)
distance ≈ 138.27 nautical miles

Now, to find how fast the distance is changing at 5 PM, we can differentiate this equation implicitly with respect to time.

2 * distance * (rate of change of distance) = 2 * (105 nautical miles * rate of change of A's position) + 2 * (90 nautical miles * rate of change of B's position)

Simplifying, we get:

2 * distance * (rate of change of distance) = 2 * (105 knots * (-21 knots/hour)) + 2 * (90 knots * 18 knots/hour)

Substituting the known values, we have:

2 * 138.27 nautical miles * (rate of change of distance) = 2 * (-2205) + 2 * 1620

Simplifying further, we get:

2 * 138.27 nautical miles * (rate of change of distance) = -4410 + 3240

2 * 138.27 nautical miles * (rate of change of distance) = -1170

Finally, solving for the rate of change of distance:

(rate of change of distance) = -1170 / (2 * 138.27) ≈ -4.23 knots per hour

Therefore, the speed at which the distance between the ships is changing at 5 PM is approximately 4.23 knots per hour.

To find the rate at which the distance between the ships is changing at 5 PM, we can use the concept of relative velocity and the formula for finding the rate of change of distance between two moving objects.

Let's break down the problem:

At noon, Ship A is 20 nautical miles due west of Ship B. This means that at the starting point, the distance between the ships is 20 nautical miles.

Ship A is sailing west at 21 knots, which means its velocity is 21 knots towards the west.

Ship B is sailing north at 18 knots, which means its velocity is 18 knots towards the north.

We are interested in finding how fast the distance between the ships is changing at 5 PM. To calculate this, we'll use the formula for the rate of change of distance:

Rate of change of distance = sqrt((velocity of A)^2 + (velocity of B)^2)

Now, let's calculate the rate of change of distance between the ships at 5 PM:

First, we need to find the positions of the ships at 5 PM.

From noon to 5 PM, there are 5 hours.

Ship A is sailing west at a speed of 21 knots for 5 hours, so its westward distance is 21 knots * 5 hours = 105 nautical miles.

Ship B is sailing north at a speed of 18 knots for 5 hours, so its northward distance is 18 knots * 5 hours = 90 nautical miles.

Now, we have the positions of the ships at 5 PM. Ship A is 105 nautical miles west of the starting point, and Ship B is 90 nautical miles north of the starting point.

Next, we can calculate the distance between the ships at 5 PM using the Pythagorean theorem.

Distance = sqrt((westward distance)^2 + (northward distance)^2)
= sqrt((105)^2 + (90)^2)
= sqrt(11,025 + 8,100)
= sqrt(19,125)
≈ 138.35 nautical miles

Finally, we can calculate the rate at which the distance between the ships is changing at 5 PM using the formula:

Rate of change of distance = sqrt((velocity of A)^2 + (velocity of B)^2)
= sqrt((21 knots)^2 + (18 knots)^2)
= sqrt(441 + 324)
= sqrt(765)
≈ 27.7 knots

Therefore, the rate at which the distance between the ships is changing at 5 PM is approximately 27.7 knots.