Designing exponential decay functions Devise an exponential...

Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point $(t=0)$ and units of time. Drug metabolism A drug is eliminated from the body at a rate of $15 \% / \mathrm{hr}$. After how many hours does the amount of drug reach $10 \%$ of the initial dose?

Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point $(t=0)$ and units of time.
Drug metabolism A drug is eliminated from the body at a rate of $15 \% / \mathrm{hr}$. After how many hours does the amount of drug reach $10 \%$ of the initial dose?

Key Concepts

Decay Rate

The decay rate is the constant percentage or fraction by which a quantity decreases per unit of time. In the context of exponential decay, it determines the steepness of the decline. A higher decay rate results in a faster reduction of the quantity, and it is integral to calculating the decay constant that appears in the exponential decay equation.

Exponential Decay Functions

Exponential decay functions model situations in which a quantity decreases at a rate proportional to its current value. These functions are typically represented by the formula A(t) = A?e^(–kt), where A? is the initial amount, k is the decay constant, and t is time. This concept is foundational in various fields for modeling processes that diminish over time in a continuous manner.

Reference Point and Units of Time

Establishing a reference point, often denoted as t = 0, defines the starting point of observation for the decay process. Clearly specifying the units of time (minutes, hours, etc.) is crucial for accurately relating the decay rate and predicting the quantity at future times. This clarity ensures that the function correctly reflects the temporal dynamics of the system being modeled.

Logarithms in Solving Exponential Equations

When solving for time in an exponential decay equation (for instance, finding when the quantity reaches a certain percentage of its initial value), logarithmic functions are used to 'undo' the exponential. By taking the logarithm of both sides of the decay equation, one can isolate the time variable and solve for it, making logarithms a critical tool in analyzing decay processes.

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