1)At noon, ship A is 120 km west of ship B. Ship A is sailing east at 25 km/h an

Question

1)At noon, ship A is 120 km west of ship B. Ship A is sailing east at 25 km/h and ship B is sailing north at 20 km/h. How fast is the distance between the ships changing at 4:00 PM? Leave answer in exact not in decimal.

2)A plane flying horizontally at an altitude of 1 mi and a speed of 460 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 4mi away from the station. (Round your answer to the nearest whole number.) mi/h

Explanation / Answer

1.

Let x = horizontal distance between ship A and ship B
At noon, x = 120 and is decreasing at rate of 20km/hr ---> dx/dt = -25
x = 120 - 25t

Let y = vertical distance between ship A and ship B
At noon, y = 0 and is increasing at rate of 15 km/hr ---> dy/dt = 20
y = 20t

At 4:00 PM (t = 4), we get:
x = 120 - 25(4) = 20
y = 20(4) = 80

Let d = distance between ship A and ship B
d² = x² + y²

Differentiate both sides with respect to t:
2d dd/dt = 2x dx/dt + 2y dy/dt
dd/dt = (x dx/dt + y dy/dt) / d
dd/dt = (x dx/dt + y dy/dt) / v(x²+y²)

dd/dt = (20*-25 + 80*20) / (20²+80²)^(0.5)
dd/dt = (-500+1600) / 10(68)^0.5
dd/dt = 1100/10(68)^0.5 = 110/(68)^0.5
dd/dt = 110/(68)^0.5

Distance between the ships is increasing at a rate of = 110/(68)^0.5

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