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Exponential functions are a type of mathematical function used to model scenarios where quantities grow or decay at a proportional rate over time. Radioactive decay is a perfect example of exponential decay. It follows a predictable pattern where the quantity of the substance decreases exponentially as time passes.
Simply put, the general form of an exponential function related to decay is described by the equation:
\[ A(t) = A_0 e^{rt} \]
Where:

• \( A(t) \) is the amount of substance remaining at time \( t \)
• \( A_0 \) is the initial quantity of the substance
• \( r \) is the decay rate (a negative number for decay situations)
• \( e \) is the base of the natural logarithm, approximately equal to 2.718

This form shows how exponential decay allows us to easily predict how much of a material remains after a certain period of time given a constant decay rate. It's also used in other contexts such as population growth, or interest calculations, but the basis remains the same - change at a steady percentage rate.

Differential equations provide a way to mathematically describe how a quantity changes over time, which makes them essential in modeling natural phenomena such as radioactive decay.
In our given exercise, we encounter the differential equation \( \frac{dA}{dt} = -0.05A \). This is a first-order separable differential equation. Here, \( \frac{dA}{dt} \) stands for the rate of change of the amount with respect to time, and \( -0.05A \) indicates that the rate of change is proportional to the current amount of radioactive substance.
To solve this, we rearrange the equation to allow integration: \[ \frac{dA}{A} = -0.05 \, dt \]By integrating both sides, we can express this relationship through natural logarithms, leading to a solution of the form \( \ln |A| = -0.05t + C \), where \( C \) is an integration constant defined by initial conditions.
Understanding and solving differential equations can help solve problems related to decay, growth, and other dynamic systems.

Calculating half-life allows us to determine how long it takes for half of a given quantity of a substance to decay. This concept is particularly important in fields like radioactive decay, pharmacokinetics, and physics.
In the case of the radioactive substance from the exercise, we use the exponential decay formula to determine the half-life, \( T_{1/2} \).
We set the equation: \[ A(t) = \frac{A_0}{2} \]
Substituting our solution for \( A(t) = 3e^{-0.05t} \) gives:

• \( 3e^{-0.05t} = \frac{3}{2} \)
• Simplifying, \( e^{-0.05t} = \frac{1}{2} \)
• Taking natural logarithms, we solve: \( -0.05t = \ln \frac{1}{2} \)
• Finally, \( t = \frac{\ln 0.5}{-0.05} \)

This yields \( t \approx 13.86 \) days, meaning in roughly 13.86 days, the substance will reduce to half of its initial amount. Mastering half-life calculations enables scientists and professionals to predict how substances diminish over time, crucial for safety and effectiveness in applications such as medical treatment and nuclear energy.