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The water acidity in a pool is considered normal when the average $\mathrm{pH}$ reading of three daily measurements is between $8.2$ and $8.5$. If the first two $\mathrm{pH}$ readings are $8.48$ and $8.35$, then find the range of $\mathrm{pH}$ value for the third reading that will result in the acidity level being normal.
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Verified Answer
We have first $\mathrm{pH}$ value $=8.48$
and scond $\mathrm{pH}$ value $=8.35$
Let third pH value be $x$.
Since, average of the three $\mathrm{pH}$ value should lie between $8.2$ and $8.5$.
$$
\begin{aligned}
&\therefore 8.2 < \frac{8.48+8.35+x}{3} < 8.5 \\
&\Rightarrow 8.2 < \frac{16.83+x}{3} < 8.5 \\
&\Rightarrow 3 \times 8.2 < 16.83+x < 8.5 \times 3 \\
&\Rightarrow 24.6-16.83 < x < 25.5-16.83 \\
&\Rightarrow 7.77 < x < 8.67
\end{aligned}
$$
Thus, third $\mathrm{pH}$ value should lie between $7.77$ and $8.67$.
and scond $\mathrm{pH}$ value $=8.35$
Let third pH value be $x$.
Since, average of the three $\mathrm{pH}$ value should lie between $8.2$ and $8.5$.
$$
\begin{aligned}
&\therefore 8.2 < \frac{8.48+8.35+x}{3} < 8.5 \\
&\Rightarrow 8.2 < \frac{16.83+x}{3} < 8.5 \\
&\Rightarrow 3 \times 8.2 < 16.83+x < 8.5 \times 3 \\
&\Rightarrow 24.6-16.83 < x < 25.5-16.83 \\
&\Rightarrow 7.77 < x < 8.67
\end{aligned}
$$
Thus, third $\mathrm{pH}$ value should lie between $7.77$ and $8.67$.
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