Basic Mathematics & Logarithm
Linear Inequalities
Grade 11
Question:
<p>In drilling world's deepest hole it was found that the temperature \(T\) in degree Celsius, \(x\) km below the earth's surface was given by \(T = 30^\circ + 25^\circ(x - 3)\), \(3 \leq x \leq 15\). At what minimum depth will the temperature be between 155°C and 205°C?</p>
Step-by-Step Solution
Key Concept: Set up the compound inequality 155 ≤ 30 + 25(x-3) ≤ 205 and solve for x to find the depth range, then identify the minimum depth from this interval.
<p><strong>Step 1: Set up the compound inequality</strong></p><p>We need 155 ≤ T ≤ 205, where T = 30 + 25(x - 3)</p><p>155 ≤ 30 + 25(x - 3) ≤ 205</p><p><strong>Step 2: Solve the left inequality</strong></p><p>155 ≤ 30 + 25(x - 3)</p><p>125 ≤ 25(x - 3)</p><p>5 ≤ x - 3</p><p>x ≥ 8</p><p><strong>Step 3: Solve the right inequality</strong></p><p>30 + 25(x - 3) ≤ 205</p><p>25(x - 3) ≤ 175</p><p>x - 3 ≤ 7</p><p>x ≤ 10</p><p><strong>Step 4: Combine results</strong></p><p>From Steps 2 and 3: 8 ≤ x ≤ 10</p><p>The valid depth range is [8, 10] km, which satisfies the constraint 3 ≤ x ≤ 15.</p><p><strong>Step 5: Find minimum depth</strong></p><p>The minimum depth in the interval [8, 10] is <strong>x = 8 km</strong></p><p>∴ Answer: <strong>8 km</strong></p>
Correct Answer: 8