SAT Math Practice Question #863 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #863

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A marine bacterial culture has an initial population of 800 and grows according to the model: \[ P(t) = 800 \cdot (2.25)^{\frac{t}{10}} \] where \( t \) is the time elapsed in hours. By what percentage does the population increase every 5 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 5: notice that \( \frac{t}{10} = \left(\frac{t}{5}\right) \cdot \frac{5}{10} \). Apply exponent power rules: \( b^{mn} = (b^m)^n \).

Elimination Framework (Hint 2)

Desmos Shortcut: In Desmos, define \( P(t) = 800(2.25)^{t/10} \). Then simply calculate \( \frac{P(5) - P(0)}{P(0)} \cdot 100 \). Desmos gives the exact percentage instantly!

Masterclass Solution & Distractor Trap Analysis

To determine the growth rate every 5 hours, evaluate the factor by which the population multiplies when \( t \) increases by 5: \[ \text{Growth Multiplier} = (2.25)^{\frac{5}{10}} = (2.25)^{\frac{1}{2}} = \sqrt{2.25} = 1.50 \] A multiplier ...

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

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