It is known that 45 % of all aluminum cans distributed will...

It is known that $45 \%$ of all aluminum cans distributed will be recycled each year. A beverage company uses 250,000 lb of aluminum cans. After recycling, the amount of aluminum, in pounds, still in use after $t$ years is given by $$N(t)=250,000(0.45)^{t}$$ (Source: The Container Recycling Institute.) a) Find $N^{\prime}(t)$ b) Interpret the meaning of $N^{\prime}(t)$

It is known that $45 \%$ of all aluminum cans distributed will be recycled each year. A beverage company uses 250,000 lb of aluminum cans. After recycling, the amount of aluminum, in pounds, still in use after $t$ years is given by
$$N(t)=250,000(0.45)^{t}$$
(Source: The Container Recycling Institute.)
a) Find $N^{\prime}(t)$
b) Interpret the meaning of $N^{\prime}(t)$

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Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. In this context, it is used to describe decay or growth processes over time, such as the diminishing amount of aluminum that remains in use after recycling. This kind of function is central for modeling scenarios where the rate of change is proportional to the current amount.

Chain Rule in Differentiation

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When differentiating an exponential function with a constant base different from e, like (0.45)^t, the chain rule allows us to multiply the derivative of the exponent with the natural logarithm of the base. This process yields the rate of change of the function with respect to time.

Rate of Change in Exponential Decay Models

The derivative of the exponential decay function indicates the instantaneous rate at which the quantity is decreasing over time. In this case, the derivative N'(t) represents how quickly the amount of aluminum in use is reduced each year. Its value, typically negative due to the decay process, quantifies the decline in aluminum can usage as a function of time.

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