Question

The water acidity in a pool is considered normal when the average pH reading of three daily measurements is between $7.2$ and $7.8$. If the first two pH readings are $7.48$ and $7.85$, find the range of pH value for the third reading that will result in the acidity level being normal?

Answer 1

Hint: For the given problem, we need to find the range of the third reading depending on the other readings and range. We will use the inequalities while reading the acidity level. On simplifying the inequality we get the required answer.

Formula used: We will construct a range for the three values and try to find the range in which the third reading can lie in.
Here, the lower limit of the range is $7.2$
And the upper limit is $7.8$
Let the readings be x, y and z.
So, the average of x, y, z should be in this range.
${\text{lower limit}} \leqslant {\text{average}} \leqslant {\text{upper limit}}$
$7.2 \leqslant \dfrac{{x + y + z}}{3} \leqslant 7.8$
Here, x= $7.48$
And y = $7.85$
Z is the third reading.

Complete step-by-step solution:
For the given sum, we will use inequality.
It is given that the question stated as the water acidity in a pool is considered normal when the average pH reading of three daily measurements is between $7.2$ and $7.8$
The first two pH readings are $7.48$ and $7.85$
Let the third reading be \[z\].
We will use the formula,
${\text{lower limit}} \leqslant {\text{average}} \leqslant {\text{upper limit}}$
This is to be used to obtain the third reading \[z\].
Here, the lower limit of the range is $7.2$
Also, the upper limit is $7.8$
Now we can write it as, $x = 7.48$, $y = 7.85$ and the third reading is \[z\].
$ \Rightarrow 7.2 \leqslant \dfrac{{x + y + z}}{3} \leqslant 7.8$
Putting the values and we get,
$ \Rightarrow 7.2 \leqslant \dfrac{{7.48 + 7.85 + z}}{3} \leqslant 7.8$
On cross multiply the values and we get,
$ \Rightarrow 7.2 \times 3 \leqslant 7.48 + 7.85 + z \leqslant 7.8 \times 3$
Let us multiply the terms and we get
$21.6 \leqslant 15.33 + z \leqslant 23.4$
On subtracting $15.33$all the side we get
$21.6 - 15.33 \leqslant z \leqslant 23.4 - 15.33$
On subtracting we get,
$6.27 \leqslant z \leqslant 8.07$

Thus the range of the third reading \[z\] will be from $6.27$ to $8.07$.

Note: Lower limit is the lower boundary of the range. No values below the lower limit shall be accepted. Upper limit is the upper boundary of the range. No values greater than the upper limit shall be accepted.
While solving this kind of problem, students should not get confused between the lower limit and upper limit as that may change the results.