Exponential Function Math Example 4

Solution

1. 1
   Colony A: $N_A = 100 \cdot 2^{t/3}$. Colony B: $N_B = 500 \cdot 2^{t/8}$.
2. 2
   Set $N_A = N_B$: $100 \cdot 2^{t/3} = 500 \cdot 2^{t/8}$. Divide both sides by 100: $2^{t/3} = 5 \cdot 2^{t/8}$.
3. 3
   Take $\log_2$: $\frac{t}{3} = \log_2 5 + \frac{t}{8}$. Solve: $\frac{t}{3} - \frac{t}{8} = \log_2 5$, so $\frac{5t}{24} = \log_2 5 \approx 2.322$, giving $t \approx \frac{24 \times 2.322}{5} \approx 11.1$ hours.
4. 4
   Colony A surpasses Colony B after approximately $11.1$ hours.