### Chemical pollution model exercises answers

Here we give answers to some of the exercises on chemical pollution in a lake..

#### Exercise 1

a.

- $W_0, \cdots ,W_4$ = 0, 1, 1.2, 1.24, 1.248.
- $E=1.25$
- $W_t = 1.25 - (0.2)^t \times 1.25 \qquad t = 0, 1, \cdots$
- $W_{100}=1.25$
- $T_{1/2} = 0.43$
- $T_{0.02} = 2.43$

c.

- $W_0, \cdots ,W_4$ = 0, 100, 120, 124, 124.8.
- $E=125$
- $W_t = 125 - (0.2)^t \times 125 \qquad t = 0, 1, \cdots$
- $W_{100}=125$
- $T_{1/2} = 0.43$
- $T_{0.02} = 2.43$

e.

- $W_0, \cdots ,W_4$ = 0, 10, 10.5, 10.525, 10.52625.
- $E= \frac{10}{0.95} \approx 10.52631$
- $W_t = 10.52631 - (0.05)^t \times 10.52631 \qquad t = 0, 1, \cdots$
- $W_{100}=10.52631$
- $T_{1/2} = 0.23$
- $T_{0.02} = 1.31$

#### Exercise 2

a.

- $W_0, \cdots ,W_4$ = 0, 1, 0.8, 0.84, 0.832..
- $E \approx 0.83333$
- $W_t = 0.83333 - (-0.2)^t \times 0.83333 \qquad t = 0, 1, \cdots$
- $W_{100} \approx 0.83333$

c.

- $W_0, \cdots ,W_4$ = 0, 100, 80, 84, 83.2.
- $E \approx 83.333$
- $W_t = 83.333 - (-0.2)^t \times 83.333 \qquad t = 0, 1, \cdots$
- $W_{100} \approx 83.333$

e.

- $W_0, \cdots ,W_4$ = 0, 10, 9.5, 9.524, 9.52375.
- $E \approx 9.25238$
- $W_t = 9.25238 - (-0.05)^t \times 9.25238 \qquad t = 0, 1, \cdots$
- $W_{100} \approx 9.25238$

#### Exercise 3

a

- $W_0, \cdots ,W_4$ = 0, 1,0 , 1, 0.
- The sequence continues to alternate between 0 and 1.

c

- $W_0, \cdots ,W_4$ = 0, 1, 2, 3, 4.
- The sequence is the sequence of non-negative integers.

e

- $W_0, \cdots ,W_4$ = 0, 1, 3, 7, 15.
- The sequence is $W_t = 2^t - 1$.

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