Chapter 5: Applications of Integration
Section 5.6: Differential Equations
A tank contains 50 lbs of salt dissolved in 1000 gallons of water. Brine containing 1/100 lb of salt per gal of water enters the tank at a rate of 40 gal per minute, mixes instantaneously, and drains at the same rate. Determine the tank's salt content 15 minutes later.
Let xt denote the amount of salt in the tank at time t, with x in pounds, and t in minutes.
The rate at which salt enters the tank is 40 gal/min times 1/100 lbs/gal, or 2/5 lb/min.
The rate at which salt leaves the tank is −401000x=−x/25 lbs/min.
The rate of change of salt in the tank, namely, dxdt or x., is then the difference between the rates at which salt enters and leaves the tank. The units would be lbs/min.
Since the tank initially contained 50 lbs of salt, the governing initial-value problem is
the solution of which is
Figure 5.6.5(a) provides a graph of xt for t∈0,15. The salt content in the tank at t=15 is
x15=10+40 e−3 t/5≐31.95 lbs
Figure 5.6.5(a) Solution of IVP
The limiting value of xt is 10 lbs, attained when each of the 1000 gallons of brine in the tank has the concentration of the incoming fluid, namely, 1/100 lb/gal. This limiting value is corroborated analytically by letting t→∞ in x⁡t=10+40⁢ⅇ−t/25. In fact, this shows x>10 for all finite t, an observation useful in a stepwise solution of the IVP by separation of variables.
The differential equation is easily put into the separated form dx25−x25=25 dx10−x=dt from which, by antidifferentiation of both sides, it follows that −25 ln(10−x)=t+c. Dividing through by −25 and exponentiating both sides leads to
In the third line, e−c/25 is an arbitrary constant A because c is. The absolute values are removed in line four by recalling that x>10 and introducing the appropriate minus sign on the right.
Finally, imposing the initial condition x0=50 gives 50=10+A so that A=40 and x=10+40 e−t/25.
Solution via Context Panel
Write the differential equation that models the amount of salt in the tank, and append to it the initial condition.
Press the Enter key.
Context Panel: Solve DE≻xt
The Context Panel can be used to select the right-hand side, and then used to invoke the Plot Builder to draw a graph akin to the one shown in Figure 5.6.5(a).
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