Chapter 8.5, Problem 54E

a.

To determine

To write: a recursive rule for the amount of chlorine in the pool at the start of the mil week.

Answer to Problem 54E

  $a_n=0.60a_{n-1}+16$ .

Explanation of Solution

Given:

The initial amount of chlorine in a swimming pool is $34\text{ounces}$ .

And the amount of chlorine is continuously added 16 ounces and 40% of the chlorine evaporates.

Calculation:

According to the question, 40% of the chlorine evaporates, 60% remain in the swimming pool. Also 16 ounces are added each year.

  $\begin{array}{l}{a}_1=34\\ a_2=\left(1-0.40\right)a_1+16\\ a_n=0.60a_{n-1}+16\end{array}$ .

Hence, a recursive rule for the amount of chlorine in the pool at the start of the mil week is $a_n=0.60a_{n-1}+16$ .

b.

To determine

To find: the amount of chlorine in the pool at the start of the third week.

Answer to Problem 54E

  $a_3=37.84$ .

Explanation of Solution

Given:

The initial amount of chlorine in a swimming pool is $34\text{ounces}$ .

And the amount of chlorine is continuously added 16 ounces and 40% of the chlorine evaporates.

Calculation:

From part (a) $a_n=0.8a_{n-1}+5000$ .

Therefore, start of the 3rd week is

  $\begin{array}{l}{a}_n=0.60a_{n-1}+16\\ a_3=0.6a_2+16=37.84\end{array}$

Hence, the number of members at the start of the 5th year is $a_3=37.84$ .

c.

To determine

To describe: what happens to the amount of chlorine in the pool over time.

Answer to Problem 54E

40.

Explanation of Solution

Given:

The initial amount of chlorine in a swimming pool is $34\text{ounces}$ .

And the amount of chlorine is continuously added 16 ounces and 40% of the chlorine evaporates.

Calculation:

From part (a)

  $\begin{array}{l}{a}_1=34\\ a_3=37.84\end{array}$ .

  $\begin{array}{l}{0.6}^{n-1}\left(34\right)\to 0\\ 16\cdot \frac{1-{0.6}^{n-1}}{1-0.6}\to 16\cdot \frac{1}{1-0.6}\\ =16\cdot \frac{1}{0.4}=16\times \frac{10}{4}=40\end{array}$

Hence, he sequence approaches to 40.