At noon, ship A is 50 nautical miles due west of ship B. Ship A is sailing west at 25 knots and ship B is sailing north at 16 knots. How fast (in knots) is the distance between the ships changing at 6 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

I assume you made a diagram.

let the time passed since noon be t hours
in that time ship A has traveled 25t n-miles , and ship B has traveled 16t n-miles
let the distance between them be d n-miles

I see a right-angled triangle with base of
50 + 25t, a height of 16t and a hypotenuse of d

d^2 = (50+25t)^2 + (16t)^2
at 6:00 pm, t = 6
d^2 = 221.847

2d(dd/dt) = 2(50+25t)(25) + 2(16t)(16)

so at 6:00 pm
dd/dt = [2(50+150)(25) + 2(96)(16)]/221.847
= 58.923

check my arithmetic

This answer isn't right either & I don't understand how you got d^2 = 221.847.

that d^2 = 221.847 was a typo

should have said d = 221.847
As you can see further down I used it as such.

Alos in my third last line of my previous post I should have divided by 2(221.847) to get
29.46 instead of 58.923

Yeah. That's better. Thanks again!

To find the speed at which the distance between the ships is changing at 6 PM, we need to calculate the rate of change of the distance between the ships.

Let's set up a coordinate system where ship A is at the origin (0,0) and ship B is at (50,0).

To find the distance between the ships at any given time, we can use the distance formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2),

where (x1, y1) are the coordinates of ship A and (x2, y2) are the coordinates of ship B.

The position of ship A can be represented by the equation x = -25t and y = 0, where t is the time in hours since noon.

The position of ship B can be represented by the equation x = 50 and y = 16t.

To find the distance between the ships, substitute the values into the distance formula:

Distance = sqrt((-25t - 50)^2 + (16t - 0)^2).

Now, to find the rate of change of the distance between the ships at 6 PM, we need to find the derivative of the distance equation with respect to time (t) and evaluate it at t = 6:

d(Distance)/dt = d(sqrt((-25t - 50)^2 + (16t - 0)^2))/dt.

To simplify the calculation, we can square the expression inside the square root:

Distance^2 = (-25t - 50)^2 + (16t - 0)^2.

Now, take the derivative of Distance^2 with respect to t using the chain rule:

d(Distance^2)/dt = 2(-25t - 50)(-25) + 2(16t)(16).

Simplifying this expression will give us the derivative:

d(Distance^2)/dt = 625t + 2500 + 512t.

Now we are left with:

d(Distance^2)/dt = 1137t + 2500.

To find the rate of change of the distance between the ships at 6 PM, substitute t = 6 into this equation:

d(Distance^2)/dt = 1137(6) + 2500.

Simplifying this will give us the final answer for the rate of change of the distance between the ships at 6 PM.