Question

Given the function ( N(t) = 2187 ⋅ (0.67)ᵗ ), determine the relationship between the elapsed time (( t )) and the total number of bears (( N(t) )) studied by ecologist Alina in Siberia.

Answer 1

The relationship between the elapsed time (( t )) and the total number of bears (( N(t) )) studied by ecologist Alina in Siberia is given by the exponential function
`\( N(t) = 2187 \cdot (0.67)^t \).`

Step-by-step explanation:

Ecologist Alina's study involves an exponential growth model represented by the function
`\( N(t) = 2187 \cdot (0.67)^t \)`. In this equation, ( N(t) ) signifies the total number of bears at time ( t ). The base of the exponent, \( 0.67 \), is less than 1, indicating a decay or decrease over time. The coefficient ( 2187 ) represents the initial population of bears when ( t = 0 ). As ( t ) increases, the exponential term ( (0.67)^t ) decreases, resulting in a diminishing bear population. This model is common in ecological studies, capturing scenarios where a population decreases over time due to factors like habitat loss or resource depletion.

The initial population of 2187 bears is multiplied by the decay factor
`\( (0.67)^t \)` to calculate ( N(t) ) at any given time. This is a standard form for exponential decay, commonly used in ecological and biological studies to understand how populations change over time. The exponent ( t ) represents the elapsed time in the study, and as ( t ) increases, ( N(t) ) decreases exponentially. Understanding the mathematical relationship between ( t ) and ( N(t) ) is crucial for ecologists like Alina to make predictions about future bear populations and implement effective conservation strategies.