Derivatives

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Ship A is 70 km west of ship B and is sailing south at the rate of 25 km/hr.ship B is sailing north at the rate of 45 km/hr.how fast is the distance between the two ships changing 2 hours later?

• Derivatives -

Make a sketch of their current position, placing B at the origin and A on the negative x-axis, 70 units from the origin
At a time of t hours, let the position of Ship A be A1 and the position of ship B be B1
Now AA1 = 25t km , and BB1 = 45t km
Join A1B1
Extend A1A upwards to C and complete the right-angled triangle,
with A1C = 25t+45t = 70t, CB1 = 70 , and hypotenuse A1B1 as the distance between the two ships.
let that distance be h

h^2 = (70t)^ + 70^2
h^2 = 4900t^2 + 4900
2h dh/dt = 9800t dt/dt = 9800t

when t = 2
h^2 = 140^2+70^2
...
h =√24500 = appr 156.525 km

at t=2
dh/dt = 9800(2)/(2√24500)
= 62.61 km/h

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