A bungee jumper is falling. The bungee cord is 100 ft long, but she is still 80
ID: 2840023 • Letter: A
Question
A bungee jumper is falling. The bungee cord is 100 ft long, but she is still 80 ft above the ground and falling at 40 ft per second. You are observing from 60 ft away on the ground. From your perspective, how fast is the angle of elevation decreasing? A bungee jumper is falling. The bungee cord is 100 ft long, but she is still 80 ft above the ground and falling at 40 ft per second. You are observing from 60 ft away on the ground. From your perspective, how fast is the angle of elevation decreasing?Explanation / Answer
(a) If L is the length of the rope and r is its radius then the Volume is V=pi L r^2 and therefore
dV/dt = 0 = pi dL/dt r^2 + pi L r dr/dt from where we get dr/dt = - (r/(2L)) dL/dt. As r=1/24 ft, L=100 ft and dL/dt = 40 ft/s we have that dr/dt = -1/120 ft/s = -1/10 inch/s = -0.1 inch/s
(b) If a is the angle, h the height of the jumper and d the horizontal distance from the point of impact (6 ft) then
tan a = h/d and therefore
sec^2 a da/dt = 1/d dh/dt or
da/dt = cos^2 a / d dh/dt.
Now, cos^2 a = d^2 / (d^2+h^2), h=80 ft, and dh/dt = - 40 ft/s so
da/dt = -0.037 rad/s = -2.14 deg/s
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