Find Solution Find Solution 7. A cup of boiling water is placed outside at 1:00

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Find Solution 7. A cup of boiling water is placed outside at 1:00 PM. One minute later the temperature of the water is 152 degree F. After another minute its temperature is 112 degree F. What is the outside temperature? 8. A tank initially contains 40 gallons of pure water. A solution with 1 gram of salt per gallon of water is added to the tank at 3 gal/min, and the resulting solution dranes out at the same rate. Find the quantity Q(t)of salt in the tank at t in r > 0. 9. A tank initially contains a solution of 10 pounds of salt in 60 gallons of water. Water with 1/2 pound of salt per gallon is added to the tank at 6 gal/min, and the resulting solution leaves at the same rate. Find the quantity Q(t) of salt in the tank at time t > 0.

Explanation / Answer

Let T(t) be the temperature at time t

Then T(t) = Ta +(T0-Ta)e-kt

Here Ta is outside temperature and T0 = initial temperature = 2120F (because boling water)

Hence T(t) = Ta+(212-Ta)e-kt

T(1) = 152 = Ta+(212-Ta)e-k => 152-Ta = 212-Ta(e-k)

Also T(2) = 112 = Ta+(212-Ta)e-2k => 112-Ta = 212-Ta(e-2k)

Dividing the two equations:

(112-Ta)/(152-Ta) = e-k

Substitute this value of e-k into the first equation, we get:

152-Ta = (212-Ta)(112-Ta)/(152-Ta)

(152-Ta)2 = (212-Ta)(112-Ta)

23104 + Ta2 - 304 Ta= 23744-324Ta +Ta2

=> -640 = -20Ta

=> 32 = Ta

Hence the outside temperature is 320F

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