At noon, ship A is 50 nautical miles due west of ship B. Ship A is sailing west at 23 knots and ship B is sailing north at 24 knots. How fast (in knots) is the distance between the ships changing at 4 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)
Make a sketch by placing B at O, the origin of the x-y plane, and A at (-50,0)
After t hours, A will be 50 + 23t nautical miles west from O, and B will be 24t nmiles north of O
Sketch the right-angled triangle and let d be the distance between them
d^2 = (50+23t)^2 + (24t)^2
2d dd/dt = 2(50+23t)(23) + 2(24t)(24
when t = 4
d^2 = 20164+9216
d = √29380
dd/dt = ( 2(50+23t)(23) + 2(24t)(24) )/(2d)
divide the right side by 2, sub in t = 4 , and d = ....
let me know what you got, so we can compare answers
To calculate the speed at which the distance between the ships is changing at 4 PM, we can use the concept of derivatives. Let's work step by step to solve the problem.
Step 1: Determine the position of ship A at 4 PM
Given that ship A is sailing west at 23 knots from the starting position at noon, we need to determine the distance it has covered from noon to 4 PM.
The time difference between noon and 4 PM is 4 hours, so ship A has traveled a distance of (23 knots * 4 hours) = 92 nautical miles west from its starting position.
Therefore, ship A is now 50 nautical miles + 92 nautical miles = 142 nautical miles due west of ship B.
Step 2: Determine the position of ship B at 4 PM
Given that ship B is sailing north at 24 knots from the starting position at noon, we need to determine the distance it has covered from noon to 4 PM.
The time difference between noon and 4 PM is 4 hours, so ship B has traveled a distance of (24 knots * 4 hours) = 96 nautical miles north from its starting position.
Step 3: Determine the distance between the ships at 4 PM
The distance between the ships at 4 PM is the distance between their positions. Using the Pythagorean theorem, we can find this distance.
Let's consider a right triangle where the hypotenuse represents the distance between the two ships. The two legs of the triangle are the distances that ship A (142 nautical miles) and ship B (96 nautical miles) have covered.
Using the Pythagorean theorem, the distance between the ships at 4 PM can be calculated as follows:
Distance^2 = (142 nautical miles)^2 + (96 nautical miles)^2
Therefore, Distance^2 = 20164 + 9216 = 29380.
Taking the square root of both sides, Distance ≈ √29380 ≈ 171.38 nautical miles.
So, at 4 PM, the distance between the ships is approximately 171.38 nautical miles.
Step 4: Calculate the rate at which the distance is changing at 4 PM
To calculate how fast the distance between the ships is changing at 4 PM, we need to find the derivative of the distance with respect to time.
Since the distance between the ships is a function of time, we can differentiate the distance equation obtained in Step 3 with respect to time.
Differentiating both sides, we get:
d(Distance)/dt = (d/dt) (√(142t)^2 + (96t)^2)
d(Distance)/dt = (1/2)*(1/√(142t)^2 + (96t)^2) * (2*(142t)*(142t) + 2*(96t)*(96t))
d(Distance)/dt = (142^2*2t + 96^2*2t) / (2√(142t)^2 + (96t)^2)
d(Distance)/dt = (2t * (142^2 + 96^2)) / (2√(142t)^2 + (96t)^2)
Substituting t = 4 (since we need to calculate the rate of change at 4 PM), we get:
d(Distance)/dt = (2*4 * (142^2 + 96^2)) / (2√(142*4)^2 + (96*4)^2)
Simplifying further, we have:
d(Distance)/dt = (8 * (20164 + 9216)) / (2√(568^2 + 96^2) + 384^2)
d(Distance)/dt = (8 * 29380) / (2√(322224 + 9216) + 147456)
d(Distance)/dt ≈ (8 * 29380) / (2√331440 + 147456)
Finally, calculating the value, we have:
d(Distance)/dt ≈ 568.36 / (727.23 + 147456)
d(Distance)/dt ≈ 568.36 / 147183.23
d(Distance)/dt ≈ 0.00386 knots
Therefore, at 4 PM, the speed at which the distance between the ships is changing is approximately 0.00386 knots.
To find the rate at which the distance between the ships is changing at 4 PM, we need to find the derivative of the distance between the ships with respect to time. Let's denote the distance between the ships as D(t), where t is the time in hours.
To determine the equation for D(t), we can use the Pythagorean theorem: D(t)^2 = (distance traveled by ship A)^2 + (distance traveled by ship B)^2.
At noon (t=0), the distance between the ships is given as 50 nautical miles. Thus, we have the condition D(0) = 50.
Let's calculate the distances traveled by each ship from noon to 4 PM (t=4):
- Ship A is sailing west at 23 knots for 4 hours, so the distance it travels is (23 knots) * (4 hours) = 92 nautical miles.
- Ship B is sailing north at 24 knots for 4 hours, so the distance it travels is (24 knots) * (4 hours) = 96 nautical miles.
Now we have the distances traveled by each ship: ship A traveled 92 nautical miles and ship B traveled 96 nautical miles. Plugging these values into the Pythagorean theorem equation, we can solve for D(4):
D(4)^2 = (92 nautical miles)^2 + (96 nautical miles)^2.
Calculating this equation will give us the square of the distance between the ships at 4 PM.
Finally, to find the rate at which the distance between the ships is changing at 4 PM, we differentiate D(t) with respect to t and evaluate it at t=4:
dD/dt = the derivative of D(t) with respect to t.
Substituting t=4 into dD/dt will give us the rate at which the distance between the ships is changing at 4 PM.