Under ideal conditions, a population of rabbits has an...

Under ideal conditions, a population of rabbits has an exponential growth rate of $11.7 \%$ per day. Consider an initial population of 100 rabbits.
a) Find the exponential growth function.
b) What will the population be after 7 days? after 2 weeks?
c) Find the doubling time.

Key Concepts

Doubling Time

Doubling time is the period required for an exponentially growing quantity to double in size. It provides an intuitive measure of the speed of growth and can be calculated using the formula T_d = (ln 2)/r for continuous growth. This concept helps in comparing growth dynamics over different time scales.

Exponential Growth

Exponential growth describes a process where the rate of change of a quantity is proportional to the current amount, leading to rapid increases over time. This concept is essential in understanding populations, such as rabbits, that grow by a constant percentage over regular time intervals.

Exponential Growth Function

An exponential growth function mathematically models situations where quantities increase at a constant rate relative to their current size. The general form is P(t) = P?e^(rt) for continuous growth or P(t) = P?(1 + r)^t for discrete intervals, where P? is the initial amount, r is the growth rate, and t is time.

Growth Rate

The growth rate is the constant percentage increase per time period in an exponential growth model. It is a crucial parameter in the model that quantifies how fast the population or quantity is increasing, and it is often expressed either as a decimal in the exponential function or as a percentage.

Time Variable in Growth Models

The time variable represents the independent component in an exponential growth model, indicating how long the process has been occurring. It is essential for calculating future values of the quantity being modeled, allowing prediction of population sizes or investment amounts at various points in time.

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