Math Insight

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

Exercise 1

a.

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

c.

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

e.

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

Exercise 2

a.

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

c.

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

e.

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

Exercise 3

a

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

c

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

e

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