Basic Mathematics & Logarithm
Linear inequalities - word problems
Grade 11
Question:
<p>The water acidity in a pool is considered normal when the average pH reading of three daily measurements is between 8.2 and 8.5. If the first two pH readings are 8.48 and 8.35, find the range of pH value for the third reading that will result in the acidity level being normal.</p>
<p>7.77 and 8.67</p>
<p>6.76 and 7.67</p>
<p>5.75 and 6.67</p>
<p>8.87 and 9.69</p>
Step-by-Step Solution
Key Concept: The average of three pH readings must satisfy 8.2 ≤ (reading₁ + reading₂ + reading₃)/3 ≤ 8.5. Multiply all parts by 3 and isolate the third reading to find its valid range.
<p><strong>Step 1:</strong> Let the third pH reading be <em>x</em>. The average of three readings must satisfy:</p><p>8.2 ≤ (8.48 + 8.35 + <em>x</em>)/3 ≤ 8.5</p><p><strong>Step 2:</strong> Multiply all parts by 3:</p><p>3(8.2) ≤ 8.48 + 8.35 + <em>x</em> ≤ 3(8.5)</p><p>24.6 ≤ 16.83 + <em>x</em> ≤ 25.5</p><p><strong>Step 3:</strong> Subtract 16.83 from all parts:</p><p>24.6 - 16.83 ≤ <em>x</em> ≤ 25.5 - 16.83</p><p>7.77 ≤ <em>x</em> ≤ 8.67</p><p><strong>Step 4:</strong> Verify: Average = (8.48 + 8.35 + 7.77)/3 = 24.6/3 = 8.2 ✓</p><p>Average = (8.48 + 8.35 + 8.67)/3 = 25.5/3 = 8.5 ✓</p><p>∴ Answer: The third pH reading must be in the range <strong>7.77 ≤ <em>x</em> ≤ 8.67</strong></p>
Correct Answer: A