Uninhibited growth can be modeled by exponential functions other than​ A(t) ​=Upper A 0 e Superscript kt. For ​ example, if an initial population Upper P 0 requires n units of time to​ triple, then the function ​P(t)equals Upper P 0 (3 )Superscript StartFraction t Over n EndFraction models the size of the population at time t. An insect population grows exponentially. Complete the parts a through d below.

The question is incomplete. Here is the complete question.

Uninhibited growth can be modeled by exponential functions other than [tex]A(t)=A_{0}e^{kt}[/tex]. for example, if an initial population P₀ requires n units of time to triple, then the function [tex]P(t)=P_{0}(3)^{\frac{t}{n} }[/tex] models the size of the population at time t. An insect population grows exponentially. Complete the parts a through d below.

a) If the population triples in 30 days, and 50 insects are present initially, write an exponential function of the form [tex]P(t)=P_{0}(3)^{\frac{t}{n} }[/tex] that models the population.

b) What will the population be in 47 days?

c) When wil the population reach 750?

d) Express the model from part (a) in the form [tex]A(t)=A_{0}e^{kt}[/tex].

Answer: a) [tex]P(t)=50(3)^{\frac{t}{30} }[/tex]

              b) P(t) = 280 insects

              c) t = 74 days

             d) [tex]A(t)=50e^{0.037t}[/tex]

Step-by-step explanation:

a) n is time necessary to triple the population of insects, i.e., n = 30 and P₀ = 50. So, Exponential equation for growth is

[tex]P(t)=50(3)^{\frac{t}{30} }[/tex]

b) In t = 47 days:

[tex]P(t)=50(3)^{\frac{t}{30} }[/tex]

[tex]P(47)=50(3)^{\frac{47}{30} }[/tex]

[tex]P(47)=50(3)^{1.567}[/tex]

P(47) = 280

In 47 days, population of insects will be 280

c) P(t) = 750

[tex]750=50(3)^{\frac{t}{30} }[/tex]

[tex]\frac{750}{50}=(3)^{\frac{t}{30} }[/tex]

[tex](3)^{\frac{t}{n} }=15[/tex]

Using the property Power Rule of logarithm:

[tex]log(3)^{\frac{t}{30} }=log15[/tex]

[tex]\frac{t}{30}log(3)=log15[/tex]

[tex]t=\frac{log15}{log3} .30[/tex]

t = 74

To reach a population of 750 insects, it will take 74 days

d) To express the population growth into the described form, determine the constant k, using the following:

A(t) = 3A₀ and t = 30

[tex]A(t)=A_{0}e^{kt}[/tex]

[tex]3A_{0}=A_{0}e^{30k}[/tex]

[tex]3=e^{30k}[/tex]

Use Power Rule again:

[tex]ln3=ln(e^{30k})[/tex]

[tex]ln3=30k[/tex]

[tex]k=\frac{ln3}{30}[/tex]

k = 0.037

Equation for exponential growth will be:

[tex]A(t)=50e^{0.037t}[/tex]

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