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an observer stands 1200 ft away from a launch pad to observe a rocket launch. Th

Calculus
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an observer stands 1200 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of 50000/a ft/sec, where a is the altitude of the rocket. Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when te rocket is 500 ft from the ground?

Apr 7th, 2015

Let x = the distance between the observer and the launch pad

Let D = the distance between the rocket and the observer

Then,

D^2 = a^2 + x^2

differentiate both sides with respect to time

2D*dD/dt = 2a*da/dt + 2x*dx/dt

since the distance between observer and launch pad doesn't change, dx/dt = 0

So then the equation reduces to

2D*dD/dt = 2a*da/dt

at 500 ft from the ground

D = sqrt(500^2 + 1200^2) = 1300 ft

and we already know that da/dt = 50,000 ft/sec

So then,

2(1300)dD/dt = 2(500)(50000)

dD/dt = (500)(50000) / 1300

= 19,230.8 ft/sec


Apr 7th, 2015

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