Find an exponential function of the form P(t)=P_0n^{frac{t}{T}} that models the situation, and then find the equivalent exponential model of the form P(t)=P_0e^{rt} 1) Doubling time of 25 weeks, initial population of 1300

Emily-Jane Bray

Emily-Jane Bray

Answered question

2021-01-27

Find an exponential function of the form P(t)=P0ntT that models the situation, and then find the equivalent exponential model of the form P(t)=P0ert
1) Doubling time of 25 weeks, initial population of 1300

Answer & Explanation

dieseisB

dieseisB

Skilled2021-01-28Added 85 answers

Given: Initial population =P0=1300 at t=0
Population =2×P0 at t=26 week
Use the given model
P(t)=P0ntT
Substitute P0=1300
P(t)=1300ntT
Substitute P(t)=2×1300 and t=25 weeks
2×1300=1300n25T
n25T=2×13001300
n25T=2
This equation holds true only if n=2 and T=25
Therefore, the possible model for the given situation is P(t)=1300(2)t25 where t is in weeks and P(t) is the population at any time t.
Since
The equation P(t)=P0ntT is equivalent to the equation P(t)=P0eet thherefore
P0ntT=P0ert
ntT=ert
Take ln both sides of the equation
ln(n)tT=lnert
tTln(n)=rt
1Tln(n)=r
Here, substitute the value if n=2 and T=25
125ln(2)=r
r=0.0277
Hence, the required model is P(t)=1300e0.0277t where t is in weeks.

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