Question

find an exponential function of the form $p(t) = p_0 n^{\frac{1}{t}t}$ that models the situation, and then find the equivalent exponential model of the form $p(t) = p_0 e^{rt}$
tripling time of 6 months, initial population of 5500

find an exponential function of the form $p(t) = p_0 n^{\frac{1}{t}t}$ that models the situation

the exponential function is $p(t) = \square$
(use integers or fractions for any numbers in the expression.)

find the equivalent exponential model of the form $p(t) = p_0 e^{rt}$

the exponential model is $p(t) = \square e^{\square t}$
(round to four decimal places as needed.)

Explanation:

Step1: Identify initial values

Initial population \( P_0 = 5500 \), tripling time \( t = 6 \) months. At \( t=6 \), \( P(6)=3P_0 \).

Step2: Find \( n \) for \( P(t)=P_0n^{t/6} \)

Substitute \( t=6 \): \( 3P_0 = P_0n^{6/6} \Rightarrow n=3 \). So \( P(t)=5500 \cdot 3^{t/6} \).

Step3: Convert to \( P(t)=P_0e^{kt} \)

Set \( 3^{t/6}=e^{kt} \). Take ln: \( \frac{t}{6}\ln3 = kt \Rightarrow k=\frac{\ln3}{6} \approx 0.1823 \).

Answer:

First function: \( 5500 \cdot 3^{t/6} \)
Second model: \( 5500e^{0.1823t} \)