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Lake Water Level Decreasing 3% Weekly: Find Level After 8 Weeks
Mathematics
High School
Question Content
Over the past several months, the water level of a lake has been decreasing by 3% each week. If the highest water level before the decrease started was 520 ft, what was the level at the end of 8 weeks?
Correct Answer
B
Detailed Solution Steps
1
Identify the exponential decay formula: \( A = P(1 - r)^t \), where \( P = 520 \), \( r = 0.03 \), \( t = 8 \)
2
Substitute values: \( A = 520(1 - 0.03)^8 = 520(0.97)^8 \)
3
Calculate \( 0.97^8 \approx 0.7837 \), then \( A \approx 520 \times 0.7837 \approx 407.5 \) ft
Knowledge Points Involved
1
Exponential Decay Formula
Used for quantities decreasing by a constant percentage rate over time: \( A = P(1 - r)^t \), where \( P \) is initial amount, \( r \) is rate, \( t \) is time.
2
Percent Change
Converting a percentage (e.g., 3%) to a decimal (0.03) for use in exponential formulas.
3
Exponentiation
Calculating powers of a decimal (e.g., \( 0.97^8 \)) to find the remaining percentage after multiple time periods.
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