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 20 knots. How fast (in knots) is the distance between the ships changing at 7 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)
A is west of B's northerly course
... distance is ... 50 + (7 * 25) = 225
B is north of A's westerly course
... distance is ... 7 * 20 = 140
the two courses form a right triangle
... with the distance (D) between the ships as the hypotenuse
... A^2 + B^2 = D^2
differentiating with respect to time (t) ... 2A dA/dt + 2B dB/dt = 2D dD/dt
dD/dt = [(225 * 25) + (140 * 20)] / √(225^2 + 140^2)
To find the rate at which the distance between the ships is changing at 7 PM, we need to find the rate of change of the distance with respect to time.
Let's assume that ship A is at coordinates (x, 0) at noon, where x is a positive number representing the distance traveled by ship A, and ship B is at coordinates (0, y), where y is a positive number representing the distance traveled by ship B.
Since ship A is sailing due west at 25 knots, its coordinates at 7 PM would be (x + 25 * (7 - 12), 0) = (x - 125, 0).
Similarly, since ship B is sailing north at 20 knots, its coordinates at 7 PM would be (0, y + 20 * (7 - 12)) = (0, y - 100).
Now, using the distance formula, we can find the distance between the two ships at 7 PM:
Distance^2 = (x - 125)^2 + (y - 100)^2
Differentiating both sides of the equation with respect to time, we get:
2 * Distance * Rate_of_change_of_distance = 2 * (x - 125) * Rate_of_change_of_x + 2 * (y - 100) * Rate_of_change_of_y
We want to find the Rate_of_change_of_distance at 7 PM, which is what we need to solve for.
Given that ship A is 50 nautical miles due west of ship B at noon, we have the initial condition:
Distance = sqrt((x - 50)^2 + y^2)
To solve for Rate_of_change_of_distance, we need to find the values of Rate_of_change_of_x, Rate_of_change_of_y, x, and y at 7 PM.
Given that ship A is sailing west at 25 knots, the Rate_of_change_of_x = -25.
Given that ship B is sailing north at 20 knots, the Rate_of_change_of_y = 20.
Given that ship A is sailing west for 7 hours, we have:
x - 125 = 25 * 7
x - 125 = 175
x = 300
Given that ship B is sailing north for 7 hours, we have:
y - 100 = 20 * 7
y - 100 = 140
y = 240
Now, we can substitute these values into our equation and solve for Rate_of_change_of_distance:
2 * Distance * Rate_of_change_of_distance = 2 * (300 - 125) * (-25) + 2 * (240 - 100) * (20)
2 * sqrt((300 - 50)^2 + 240^2) * Rate_of_change_of_distance = 2 * (175) * (-25) + 2 * (140) * (20)
2 * sqrt(250^2 + 240^2) * Rate_of_change_of_distance = 2 * 175 * (-25) + 2 * 140 * 20
2 * sqrt(250^2 + 240^2) * Rate_of_change_of_distance = -8750 + 5600
2 * sqrt(250^2 + 240^2) * Rate_of_change_of_distance = -3150
Rate_of_change_of_distance = -3150 / (2 * sqrt(250^2 + 240^2))
Rate_of_change_of_distance = -3150 / (2 * sqrt(62500 + 57600))
Rate_of_change_of_distance = -3150 / (2 * sqrt(120100))
Rate_of_change_of_distance = -3150 / (2 * 346.88)
Rate_of_change_of_distance = -3150 / 693.76
Rate_of_change_of_distance ≈ -4.53 knots
Therefore, the rate at which the distance between the ships is changing at 7 PM is approximately -4.53 knots.
To find the speed at which the distance between the ships is changing at 7 PM, we need to find the rate of change of the distance between the ships with respect to time.
Let's call the distance between the ships at noon as D(m), where "m" represents noon.
First, we need to find the distance traveled by each ship from noon to 7 PM.
Ship A
The ship A is traveling west at 25 knots. From noon to 7 PM, which is a duration of 7 hours, ship A will travel (25 knots * 7 hours) = 175 nautical miles.
Ship B
The ship B is traveling north at 20 knots. From noon to 7 PM, ship B will travel (20 knots * 7 hours) = 140 nautical miles.
Now, we can find the distance between the ships at 7 PM using the Pythagorean theorem.
The horizontal distance traveled by ship A = 175 nautical miles
The vertical distance traveled by ship B = 140 nautical miles
Using the Pythagorean theorem, we can find the distance between the ships at 7 PM:
D^2 = (175 nautical miles)^2 + (140 nautical miles)^2
Simplifying the equation, we get:
D^2 = 30625 + 19600
D^2 = 50225
D ≈ 224.21 nautical miles
Now, to find the rate of change of the distance between the ships at 7 PM, we need to differentiate D with respect to time (t) and find the value at t = 7.
dD/dt represents the rate of change of D with respect to t.
Differentiating D^2 = 50225 with respect to t, we get:
2D(dD/dt) = 0
Simplifying the equation, we get:
dD/dt = 0 / (2 * 224.21)
dD/dt = 0 knots
Therefore, at 7 PM, the speed at which the distance between the ships is changing is 0 knots.