Question
equation for constant growth rate
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GPT-4o mini
A constant growth rate can typically be described using an exponential growth model. The general equation for this model is:
\[
P(t) = P_0 \cdot e^{rt}
\]
Where:
- \(P(t)\) is the population (or quantity) at time \(t\),
- \(P_0\) is the initial quantity (at time \(t = 0\)),
- \(r\) is the constant growth rate (expressed as a decimal),
- \(t\) is the time,
- \(e\) is the base of the natural logarithm (approximately equal to 2.71828).
If you're dealing with discrete time intervals (e.g., years), the equation can also be expressed in a simpler form:
\[
P(t) = P_0 \cdot (1 + r)^t
\]
This version is often used when the growth is compounded at specific intervals, like annually. In this case, \(r\) is still the growth rate, expressed as a decimal (e.g., a growth rate of 5% would be \(r = 0.05\)).
Both equations describe the same underlying principle of constant exponential growth, but they are used in slightly different contexts.
\[
P(t) = P_0 \cdot e^{rt}
\]
Where:
- \(P(t)\) is the population (or quantity) at time \(t\),
- \(P_0\) is the initial quantity (at time \(t = 0\)),
- \(r\) is the constant growth rate (expressed as a decimal),
- \(t\) is the time,
- \(e\) is the base of the natural logarithm (approximately equal to 2.71828).
If you're dealing with discrete time intervals (e.g., years), the equation can also be expressed in a simpler form:
\[
P(t) = P_0 \cdot (1 + r)^t
\]
This version is often used when the growth is compounded at specific intervals, like annually. In this case, \(r\) is still the growth rate, expressed as a decimal (e.g., a growth rate of 5% would be \(r = 0.05\)).
Both equations describe the same underlying principle of constant exponential growth, but they are used in slightly different contexts.
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