Repeat Exercise 8, trying other functions of t in place of f(t) such as t³. Find which functions give a recognizable answer, and verify that answer using Equation (1).

Data from Exercises 8

Repeat Exercise 7, but in place of f(t) put a * e Λ (–p * t), the form of P_L(t) used in Exercises 5. Verify that the solution is equivalent to the solution given in Exercise 6.

Data from Exercises 7

At the website Wolfram Alpha.com, you can enter “y´(t) = (f(t) – y) * k, y(0) = a” to solve the initial value problem in this Extended Application, where we have used y(t) to represent P_L(t), f(t) to represent P_i(t), and a to represent P_L(0) Try this, and verify that the solution is equivalent to Equation (1).

Equation 1

Data from Exercises 5

Suppose that instead of assuming that all pollution inflow immediately ceases, we model P_i(t) by a decaying exponential of the form a · e^–pt where p is a constant that tells us how fast the inflow is being cleaned of pollution. To simplify things, we’ll also assume that initially the inflow and the lake have the same pollution concentration, so a = P_L(0)Now substitute a = P_L(0)e^–px for P_i(x) in Equation (1), and evaluate the integral as a function of t.

Data from Exercises 6.

When you simplify the right-hand side of Equation (1) using your new expression for the integral, and then factor out and divide by P_L(0)you’ll get the following nice expression for the ratio P_L(t)/P_L(0):

Transcribed Image Text:

e-k[P₂(0) + k["P,(x)e²* dx] PL(t)= е (1)