SAT Math Practice Question #864 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #864

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A marine bacterial culture has an initial population of 1200 and grows according to the model: \[ P(t) = 1200 \cdot (2.56)^{\frac{t}{12}} \] where \( t \) is the time elapsed in hours. By what percentage does the population increase every 6 hours?
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Socratic AI Engine Step-by-Step Derivation
0ms Precomputed
Tactical Insight (Hint 1)

Rewrite the exponent to isolate time increments of 6: notice that \( \frac{t}{12} = \left(\frac{t}{6}\right) \cdot \frac{6}{12} \). Apply exponent power rules: \( b^{mn} = (b^m)^n \).

Elimination Framework (Hint 2)

Desmos Shortcut: In Desmos, define \( P(t) = 1200(2.56)^{t/12} \). Then simply calculate \( \frac{P(6) - P(0)}{P(0)} \cdot 100 \). Desmos gives the exact percentage instantly!

Masterclass Solution & Distractor Trap Analysis

To determine the growth rate every 6 hours, evaluate the factor by which the population multiplies when \( t \) increases by 6: \[ \text{Growth Multiplier} = (2.56)^{\frac{6}{12}} = (2.56)^{\frac{1}{2}} = \sqrt{2.56} = 1.60 \] A multiplier ...

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

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