A man starts walking north at 3 ft/s from a point P. Five minutes later a woman

Question

A man starts walking north at 3 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 min after the woman starts walking? A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 7 m from the dock? Water is leaking out of an inverted conical tank at a rate of 12,500 cm^3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank.

Explanation / Answer

(1)Here, x-distance = 500ft
y-distance travelled by man in 5 min = 5*60*3 = 900 ft
y-distance travelled at time t secs after woman starts = 900+3t+5t = 900+8t
by Pythagoras, distance z between them at time t, z = (500² + (900+8t)²)
dz/dt = 8*(8t+900))/(500² + (900+8t)²)
at t =15 min= 900 sec

So, dz/dt =8*(8100)/(500² + (8100)²)

7.984 ft/s

(2)Let  y is the length of the rope and x is the distance of the boat from the dock
y²=1²+x²
when x=7, y=52
2*y*dy/dt=2*x*dx/dt
dy/dt= 1 m/s
Solve for dx/dt when x=7 m/s, y=52, and dy/dt= 1 m/s
so, dx/dt=52/7=1.01 m/s
The boat is approaching the dock at 1.01 m/s

(3)dV/dt = dIn/dt - dOut/ dt

=> dIn/dt=dV/dt +dOut/dt
dOut/dt = 12500cm^3/min
d=400 cm , H=600 cm
------d-----------
............../
....._r_/
    .h|../ dh/dt = 20 cm/min, h=200 cm
     .|./
      /          
To find
dIn/dt when h = 200 cm

Now by the rpoperty of similar triangle

r/200 = h/600
=> r = h/3
Now
V = (1/3)r²h
V = (1/3)(h/3)² h
V = (/27)h^3
dV/dt = (/9)h² dh/dt
dIn/dt =(/9)h² dh/dt + dOut/dt
dIn/dt (h=200 cm) = (/9) *200² *20 + 12500
dIn/dt (h=200 cm) 291752.68 cm^3/min

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