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Exponential functions are a fundamental concept in calculus and are frequently applied in various fields, including business. These functions have the form \(f(t) = ab^t\), where \(a\) is the initial value, \(b\) is the base or growth factor, and \(t\) is the exponent, often representing time. In our problem, the function \(e^{-0.01t}\) is an exponential function with base \(e\), the natural exponential constant approximately equal to 2.718. Exponential functions are characterized by their consistent multiplicative rate of change.

In practical business applications, these functions help model behaviors where change happens proportionally to the present amount, such as decay mentioned in our bulb lifetime problem. Understanding how these functions work is crucial for making predictions and decisions based on logistic growth or decay trends in real-world scenarios.

Decay models are specific types of exponential functions that describe processes of decrease or reduction over time, often following a negative exponential function like \(e^{-kt}\). In the case of our light bulb problem, we use a decay model to determine how the proportion of bulbs decreases over time. Here, the decay rate is represented by the term \(-0.01t\), indicating that every week, the proportion of functioning bulbs decreases exponentially by a factor of \(0.01t\).

Important aspects of decay models include:

• **Decay constant (\(k\))**: Determines how quickly the value decays. A larger \(k\) means faster decay.
• **Initial quantity**: The starting point that decays over time, often assumed to be 100% at \(t=0\).
• **Continuous decay**: Unlike linear models, decay models assume change is continuous, making them more realistic for many natural and business phenomena.

Decay models are invaluable in business for understanding product lifespan, depreciation, and resource depletion.

When tackling exponential decay problems in business, a structured approach helps ensure accuracy and clarity. Following the problem-solving steps as demonstrated in the light bulb exercise can improve understanding and efficiency:

• **Step 1: Identify the Given Function**: Recognize the exponential function provided. In our case, it is \(e^{-0.01t}\).
• **Step 2: Substitute the Given Time**: Place the specific time value into the function to evaluate a particular scenario, like finding the proportion of bulbs lasting more than 10 weeks.
• **Step 3: Simplify the Exponent**: Perform any necessary arithmetic operations to simplify the exponent. This reduces potential errors in calculating the function's outcome.
• **Step 4: Calculate the Exponential Value**: Use a calculator or computational tool to find the value of the exponential function, ensuring precision in your result.

Following these problem-solving strategies fosters careful analysis and correct application of mathematical principles to business problems.