Continuously compounded interest and exponential growth are closely related concepts because they both involve the mathematical idea of growth at a rate proportional to the current value.
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Mathematical Relationship: Continuously compounded interest can be described by the formula \( A = Pe^{rt} \), where:
- \( A \) is the amount of money accumulated after time \( t \),
- \( P \) is the principal amount (the initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years, and
- \( e \) is Euler's number (approximately equal to 2.71828).
This formula demonstrates that the amount of money grows exponentially over time due to the continuous nature of the compounding.
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Exponential Growth Model: Exponential growth occurs when a quantity increases at a rate proportional to its current value. This can be modeled generally with the equation \( y(t) = y_0 e^{kt} \), where \( y_0 \) is the initial quantity and \( k \) is the growth constant. In the context of continuously compounded interest, the growth of the investment over time reflects this exponential growth behavior, where the investment grows faster as it increases, due to continuous reinvestment of interest.
In summary, both concepts demonstrate how quantities can grow rapidly over time under the influence of rates that scale with their current values, characterized mathematically by the exponential function.