Chapter 3.6, Problem 42PPE

To determine

To calculate: the possible value of the water in a swimming pool for the third reading $p$ which will make the average pH normal.

Answer to Problem 42PPE

The possible value for the third reading p which will make the average pH normal is $6.6\le p\le \mathrm{8.4.}$

Explanation of Solution

Given information:

The average of the three pH readings is between 7.2 and 7.8, inclusive.

The first two readings for a swimming pool are 7.4 and 7.9.

Calculation:

The first two readings for a swimming pool are 7.4 and 7.9.

Let $p$ denote the third reading for the swimming pool, then the average of the three pH readings is $\frac{7.4+7.6+p}{3}$

But it is given that the average of the three pH readings is between 7.2 and 7.8, inclusive.

Then the inequality that represents the average of the three pH readings is

  $7.2\le \frac{7.4+7.6+p}{3}\le 7.8$

Solve the obtained compound inequality by working on all three parts of the inequality at the same time. Isolate the variable between the inequality symbols.

  $\begin{array}{c}7.2\le \frac{7.4+7.6+p}{3}\le 7.8\\ 21.6\le 7.4+7.6+p\le 23.4\end{array}$

  $21.6-15\le 7.4+7.6+p-15\le 23.4-15$

  $6.6\le p\le 8.4$

Therefore, the possible value for the third reading $p$ which will make the average pH normal is $6.6\le p\le 8.4$ .