The graph shows the deer population in a Pennsylvania county...

The graph shows the deer population in a Pennsylvania county between 1996 and 2000. Assume that the population grows exponentially. (a) What was the deer population in 1996? (b) Find a function that models the deer population $t$ years after 1996. (c) What is the projected deer population in 2004? (d) In what year will the deer population reach 100,000?

The graph shows the deer population in a Pennsylvania county between 1996 and 2000. Assume that the population grows exponentially.
(a) What was the deer population in 1996?
(b) Find a function that models the deer population $t$ years after 1996.
(c) What is the projected deer population in 2004?
(d) In what year will the deer population reach 100,000?

Key Concepts

Solving Exponential Equations Using Logarithms

To determine the time required for an exponentially growing population to reach a certain level, logarithms are used to invert the exponential function. By taking the logarithm of both sides of the equation, one can solve for the time variable, making it possible to predict when a target population size will be achieved.

Exponential Growth

Exponential growth describes a process in which the quantity increases by the same proportional rate over successive equal time intervals. This is modeled mathematically where the rate of change of the population is directly proportional to the current amount, leading to faster increases as the population grows.

Exponential Function Model

An exponential function model is commonly written in the form P(t) = P(0) * e^(kt) or P(t) = P(0) * a^t, where P(0) is the initial population, k is the continuous growth rate, and t represents time. This model is used to describe processes like population growth, where changes occur continually over time.

Initial Value

The initial value in any exponential growth model is the starting quantity at time zero. In the context of population modeling, it represents the population size at the beginning of the observation period, providing a baseline from which growth is measured.

Growth Rate

The growth rate is the constant factor by which the population increases over each time interval in an exponential growth scenario. It is a critical parameter in the exponential function model, as it determines how rapidly the quantity grows over time.

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