IB Mathematics AHL 1.9 Laws of logarithms AI HL Paper 3- Exam Style Questions- New Syllabus
▶️ Answer/Explanation
a
Method 1: Use \( N = 1, 2 \)
For \( N = 1 \): \( T = a \cdot 1^b = a \implies a = 420.65 \)
For \( N = 2 \): \( 390.94 = 420.65 \cdot 2^b \implies 2^b = \frac{390.94}{420.65} \approx 0.929355 \)
\( b = \frac{\ln(0.929355)}{\ln(2)} \approx -0.10567 \)
Method 2: Use \( N = 1, 8 \)
For \( N = 1 \): \( a = 420.65 \)
For \( N = 8 \): \( 342.08 = 420.65 \cdot 8^b \implies 8^b \approx 0.813054 \)
\( b = \frac{\ln(0.813054)}{\ln(8)} \approx -0.10629 \)
Prefer Method 1 for exact match
Result: \( a = 420.65 \), \( b = -0.10567 \) [4].
b
Method 1: Direct Substitution
\( T = 420.65 \cdot 8^{-0.10567} \approx 420.65 \cdot 0.802589 \approx 337.611 \approx 337.61 \)
Method 2: Logarithmic Form
\( \ln T = \ln(420.65) – 0.10567 \cdot \ln(8) \approx 6.041885 – 0.219735 \approx 5.822150 \)
\( T \approx e^{5.822150} \approx 337.611 \approx 337.61 \)
Result: 337.67 seconds [1].
c
Method 1: Standard Formula
Error: \( |342.08 – 337.67| = 4.41 \)
Percentage error: \( \frac{4.41}{342.08} \times 100 \approx 1.29\% \)
Method 2: Direct Calculation
Error: \( 342.08 – 337.67 = 4.41 \)
Percentage: \( \frac{4.41}{342.08} \times 100 \approx 1.29\% \)
Result: 1.29% [1].
d
Model \( T = 420.65 \cdot N^{-0.10567} \) decreases as \( N \) increases
Near \( N = 8 \), model fits well (1.29% error)
For large \( N \), \( T \) approaches 0, which is unrealistic
Result: Model is valid near \( N = 8 \), but not for large \( N \) due to unrealistically low times [2].