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.