A worker accidentally spills some contaminant into a drum filled to the top with

Question

A worker accidentally spills some contaminant into a drum filled to the top with 10 gallons of water. He quickly grabs a hose and begins to pour in water to clear out the contaminant. The water flows through the hose at 2 gal/min. After ten minutes, his supervisor catches him and stops him. the worker claims he only spilled a small amount. The supervisor takes a reading and finds the concentration of the contaminant is 0.01lb/gal. How much did the worker spill?

answer 0.74lb

Explanation / Answer

suppose the rate of change with of the quantity (Q) of contaminant with respect to time (t) is dQ/dt this is described by the difference of flow in and flow out i.e. dQ/dt = rate in - rate out because the rate in is zero (not adding any contaminant) and the rate in(water) is the same as the rate out dQ/dt=-2gal/min but our units, however, do not match there is some quantity (Q) in 10 gals water or Q(t)/10gal=1 (a conversion factor) so dQ/dt=-(2gal/min)(Q/10gal) or dQ/dt=-Q/5 and 1/Q dQ=-1/5 --> lnQ=(-1/5)t -->Q=Cexp(-t/5) we also know Q(10)=0.1 (from 10 gal of 0.01 lb/gal) Q(10)=1/10=Cexp(-10/5) C=exp(2)/10=0.74 C is also the initial value

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