More Americans are buying organic fruit and vegetables and...

More Americans are buying organic fruit and vegetables and products made with organic ingredients. The amount $A(t)$, in billions of dollars, spent on organic food and beverages $t$ years after 1995 can be approximated by $$ A(t)=2.43 e^{0.18 t} $$ a) Estimate the amount that Americans spent on organic food and beverages in $2009 .$ b) Estimate the rate at which spending on organic food and beverages was growing in $2006 .$

More Americans are buying organic fruit and vegetables and products made with organic ingredients. The amount $A(t)$, in billions of dollars, spent on organic food and beverages $t$ years after 1995 can be approximated by
$$
A(t)=2.43 e^{0.18 t}
$$
a) Estimate the amount that Americans spent on organic food and beverages in $2009 .$
b) Estimate the rate at which spending on organic food and beverages was growing in $2006 .$

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Key Concepts

Function Evaluation

Function evaluation is the method of determining the value of a function at a specific input value. In the context of time-dependent models such as in economics, it is used to calculate predictions or estimates at particular time points by substituting the designated time into the model's equation.

Exponential Growth

Exponential growth describes a process where the increase in a quantity is proportional to its current value, resulting in growth that accelerates over time. This concept is vital in various disciplines such as finance, biology, and economics, and it is typically modeled using exponential functions, which illustrate how small changes initially can lead to large variations over time.

Differentiation of Exponential Functions

Differentiation is a fundamental calculus concept used to determine the rate of change of a function with respect to a variable. For exponential functions, the derivative reveals that the rate of change is proportional to the function’s current value, which highlights the self-reinforcing nature of exponential growth and is essential in understanding how quickly such processes evolve over time.

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