91Ó°ÊÓ

Exponential decay is a concept where the quantity of something decreases at a rate proportional to its current value. In our oil well scenario, this concept is important because the rate at which the oil is pumped out decreases as there is less oil left. The smaller the amount left, the slower the rate at which it reduces.
This is mathematically represented by the differential equation \( \frac{dQ}{dt} = -kQ \), where \( Q \) is the amount of oil left, \( t \) is time, and \( k \) is a positive constant known as the proportionality constant. The minus sign indicates a decrease or decay in the amount.
Once you break down this equation, it leads us to a solution of the form: \( Q(t) = Ce^{-kt} \). Here, \( C \) is determined using initial conditions, and it serves as a starting point for modeling exponential decay.

Initial conditions are the known values at the start of the scenario that are crucial for finding particular solutions to differential equations. They provide the specific circumstances of the problem at \( t = 0 \).
In our exercise, the initial condition is that there were 1 million barrels of oil in the well initially. This helps us find the constant \( C \) in the equation \( Q(t) = Ce^{-kt} \). By substituting \( t = 0 \) and \( Q(0) = 1000000 \), we calculate \( C = 1000000 \).
With this initial condition, we have the specific exponential decay model \( Q(t) = 1000000 e^{-kt} \), giving a precise formula that includes all the known conditions at the beginning.

The proportionality constant \( k \) is a pivotal element in differential equations that depicts how fast or slow the exponential decay proceeds. It essentially determines the steepness of the exponential decay curve.
In the example, we know that 500,000 barrels remain after 6 years, which allows us to determine \( k \). By substituting \( Q(6) = 500000 \) into the equation \( Q(t) = 1000000 e^{-kt} \), we find \( e^{-6k} = 0.5 \). Solving this gives \( k = -\frac{\ln(0.5)}{6} \).
Using this calculated value of \( k \), we can determine important facts like the rate of decrease when there are 600,000 barrels remaining and the time duration before only 50,000 barrels are left. This constant is crucial for understanding and predicting how rapidly a given process or phenomenon will change over time.