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Zorluk: KolayLinear and Exponential Growth

A study by a wildlife biologist shows that the population of a certain species of songbird in a state park increases by 8%8\% of its previous year's population each year. The initial population of the songbirds in the park was 150150. If tt represents the number of years since the study began, which of the following functions best models the population of songbirds, P(t)P(t), after tt years?

  1. A
    P(t)=150+12tP(t) = 150 + 12t
  2. B
    P(t)=162tP(t) = 162^t
  3. P(t)=150(1.08)tP(t) = 150(1.08)^tCevap
  4. D
    P(t)=8+150tP(t) = 8 + 150t

Cevap

The correct function is P(t)=150(1.08)tP(t) = 150(1.08)^t.
The correct answer is the function showing the initial value of 150150 multiplied by the growth factor 1.081.08 raised to the power of tt. Since the population increases by a constant percentage (8%8\%) each year, the growth is exponential. The general form of an exponential growth model is P(t)=P0(1+r)tP(t) = P_0(1 + r)^t, where P0=150P_0 = 150 is the initial value and r=0.08r = 0.08 is the growth rate. Substituting these values gives P(t)=150(1.08)tP(t) = 150(1.08)^t.

Adım Adım Çözüm

1
Identify the initial value of the population.
The initial population of songbirds is 150150.
The initial value represents the population when t=0t = 0, which corresponds to the coefficient P0P_0 in the exponential growth model.
2
Determine the growth rate and calculate the growth factor.
The population increases by 8%8\% each year, so the growth rate r=0.08r = 0.08, and the growth factor is 1+r=1+0.08=1.081 + r = 1 + 0.08 = 1.08.
An increase by a constant percentage each year represents exponential growth, where the growth factor is 1+r1 + r.
3
Formulate the exponential growth function.
P(t)=150(1.08)tP(t) = 150(1.08)^t
An exponential growth model is written in the form P(t)=P0(b)tP(t) = P_0(b)^t, where P0P_0 is the initial value and bb is the growth factor.

Anahtar Kavram

Distinguishing between linear and exponential growth, and constructing exponential models from a percent growth rate.
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Konular

Linear and Exponential Growth

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