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Difficulty: EasyExponential Functions and Equations

A wildlife biologist models the population of a certain bird species in a nature reserve using the function P(t)=120(1.05)tP(t) = 120(1.05)^t, where P(t)P(t) represents the number of birds tt years after the study began. What does the number 120120 represent in this model?

  1. The initial number of birds in the reserve when the study beganAnswer
  2. B
    The number of birds added to the reserve each year
  3. C
    The percent by which the bird population increases each year
  4. D
    The number of years it takes for the bird population to double

Answer

The initial number of birds in the reserve when the study began
In the exponential function model P(t)=a⋅btP(t) = a \cdot b^t, the coefficient aa is the value of the function at t=0t = 0. Evaluating P(0)P(0) yields 120(1.05)0=120120(1.05)^0 = 120, which corresponds to the initial number of birds in the reserve when the study began.

Step-by-Step Solution

1
Identify the standard form of the exponential growth function
P(t)=a⋅btP(t) = a \cdot b^t, where aa represents the initial value and bb represents the growth factor.
This helps map the given constants to their mathematical meanings.
2
Substitute t=0t = 0 to find the starting population
P(0)=120(1.05)0=120⋅1=120P(0) = 120(1.05)^0 = 120 \cdot 1 = 120.
Finding the value at t=0t = 0 defines the initial state of the model.
3
Interpret the initial value in the context of the problem
The number 120120 represents the number of birds present at the start of the study.
This links the mathematical result to the real-world scenario.

Key Concept

Interpreting components of an exponential function in context
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Topics

Exponential Functions and Equations

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