   Chapter 12, Problem 49RE ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Chemical mixture A 300-gal tank initially contains a solution with 100 lb of a chemical. A mixture containing 2 lb/gal of the chemical enters the tank at 3 gal/min, and the well-stirred mixture leaves at the same rate. Find an equation that gives the amount of the chemical in the tank as a function of time. How long will it be before there is 500 lb of chemical in the tank?

To determine

To calculate: The time requires to fill the tank up to 500lb of chemical when the rate at which the chemical enter and leaves the tank is 3 gal/min.

Explanation

Given Information:

The rate at which the chemical enters and leaves the tank is 3 gal/min and a 300-gal tank contains initially 100 lb solution of chemical.

Formula used:

The logarithmic rule of integrals, 1xdx=ln|x|+C where x0.

The natural logarithm property,

logab=yb=ay.

Calculation:

Consider the rate at which the chemical enters and leaves the tank is 3 gal/min.

Now, let x represent the amount of chemical in the tank, then according to the provided information, the differential equation that represents the situation is,

dxdt=323x300=6x100=600x100

Rewrite the above differential equation,

dx600x=dt100

Now, use the logarithmic rule of integrals to get,

dx600x=dt100ln|600x|=t100+C1ln|600x|=t100+C</

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