SAT Math Practice Question #861 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #861

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A marine bacterial culture has an initial population of 500 and grows according to the model: \[ P(t) = 500 \cdot (1.69)^{\frac{t}{6}} \] where \( t \) is the time elapsed in hours. By what percentage does the population increase every 3 hours?
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Socratic AI Engine Step-by-Step Derivation
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Tactical Insight (Hint 1)

Rewrite the exponent to isolate time increments of 3: notice that \( \frac{t}{6} = \left(\frac{t}{3}\right) \cdot \frac{3}{6} \). Apply exponent power rules: \( b^{mn} = (b^m)^n \).

Elimination Framework (Hint 2)

Desmos Shortcut: In Desmos, define \( P(t) = 500(1.69)^{t/6} \). Then simply calculate \( \frac{P(3) - P(0)}{P(0)} \cdot 100 \). Desmos gives the exact percentage instantly!

Masterclass Solution & Distractor Trap Analysis

To determine the growth rate every 3 hours, evaluate the factor by which the population multiplies when \( t \) increases by 3: \[ \text{Growth Multiplier} = (1.69)^{\frac{3}{6}} = (1.69)^{\frac{1}{2}} = \sqrt{1.69} = 1.30 \] A multiplier o...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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