At noon, ship A is 20 nautical miles due west of ship B. Ship A is sailing west at 23 knots and ship B is sailing north at 17 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.)

draw the diagram.

I see this

d= sqrt(ND^2+(WD+20)^2) where ND is north distance, WD is west distance
take the derivative

d'= 1/2 *1/(sqrt( ) * 2ND*ND'+2(WD+20)(WD')

ND= 17kt/hr*5hrs ND'=17kts/hr
WD=23*5 WD'=23

have fun.

what do you mean by 1/(sqrt()?

what do you mean by 1/(sqrt()?

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 and apply the Pythagorean Theorem.

Given:
- Ship A is sailing west at 23 knots.
- Ship B is sailing north at 17 knots.
- At noon, ship A is 20 nautical miles due west of ship B.

Let's break down the problem step by step:

1. Determine the positions of ships A and B at 5 PM:
- Ship A has been sailing west for 5 hours at a speed of 23 knots. Therefore, it has traveled a distance of 5 hours * 23 knots = 115 nautical miles due west from its starting point at noon.
- Ship B has been sailing north for 5 hours at a speed of 17 knots. Therefore, it has traveled a distance of 5 hours * 17 knots = 85 nautical miles due north from its starting point at noon.

2. Calculate the distance between the ships at 5 PM using the Pythagorean Theorem:
- The distance between ships A and B is the hypotenuse of a right triangle, with one side being the distance traveled by ship A (115 nautical miles) and the other side being the distance traveled by ship B (85 nautical miles).
- Using the Pythagorean Theorem, the distance between the ships at 5 PM can be found as follows:
Distance = sqrt((115)^2 + (85)^2)
= sqrt(13225 + 7225)
= sqrt(20450)
≈ 143 nautical miles.

3. Determine the speed at which the distance between the ships is changing at 5 PM:
- To find the speed at which the distance between the ships is changing, we need to calculate the derivative of the distance with respect to time.
- The distance between the ships is a function of time, given by the Pythagorean Theorem. Taking the derivative of this equation will give us the rate of change of the distance with respect to time.
- Differentiating the equation, we get:
d(Distance)/dt = (115 * d(115)/dt + 85 * d(85)/dt) / sqrt((115)^2 + (85)^2)
= (115 * 23 + 85 * 17) / sqrt((115)^2 + (85)^2)
= (2645 + 1445) / sqrt(13225 + 7225)
= 4090 / sqrt(20450)
≈ 288.77 knots.

Therefore, the speed at which the distance between the ships is changing at 5 PM is approximately 288.77 knots.