A tank contains 100 grams of a substance dissolved in a large amount of water. T

Question

A tank contains 100 grams of a substance dissolved in a large amount of water. The tank is filtered in such a way that water drains from the tank, leaving the substance behind in the tank. Consider the volume of the dissolved substance to be negligible. At what rate is the concentration (grams/liter) of the substance changing with respect to time in each scenario? the rate after 5 hours, if the tank contains 50 L of water initially, and drains at a constant rate or A L/hr? g/L/hr the rate at the instant when 20 liters remain, if the water is draining at 2.2 L/hr at that instant g/L/hr the rate in scenario (b), if the unknown substance is also being added at a rate of 30 g/hr (and there are 100 grams in the tank at that instant) g/L/hr A ladder 10 meters long rests on horizontal ground and leans against a vertical wall. The foot of the ladder is pulled away from the wall at the rate of 0.3 m/sec. How fast is the top sliding down the wall when the foot or the ladder is 6 m from the wall? m/sec

Explanation / Answer

a...Let K = concentration
K = 100 / V

. . . V = 50 - 4t

K = 100 / (50 - 4t)

dK/dt = 100 / (25-2 t)^2

. . . t = 5

dK/dt = 100 / (25 - 2*5)^2 = 4/9 grams per liter per hour

b.. K = 100 / (20 - 2.2 t)

dK/dt = 220 / (20 - 2.2 t)^2

. . . t = 0

dK/dt = 220 / (20 - 0)^2 = 11/20 grams per liter per hour

c K = (100 + 30t) / (20 - 2.2t)

dK/dt (at t=0 ) =2.04999369   grams per liter per hour

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