91Ó°ÊÓ

The continuous growth model is a key concept in mathematics and economics, used to describe how quantities such as populations, investments, and, in this case, oil consumption, grow over time continuously and at a constant percentage rate.

When we talk about oil consumption in the exercise, we're referring to a situation where the rate of consumption does not simply increase by the same amount each year, but rather, the increase is proportional to the current amount—this is the heart of continuous growth. In the given problem, the oil consumption rate increases by 1.5% each year, so the amount of oil consumed grows exponentially, a concept we'll explore further in the next section.

To calculate the total quantity after a certain period, the continuous growth model uses the formula:

\[ Quantity(t) = Quantity_0 \cdot e^{rt} \]

where:

• \(Quantity_0\) is the initial quantity,
• \(e\) is Euler's number (approximately 2.71828),
• \(r\) is the growth rate (expressed as a decimal), and
• \(t\) is the time that has passed.

This formula allows us to project the future rate of oil consumption based on the initial rate and the growth rate.

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In the context of the oil consumption problem, the exponential function represents the rate at which oil consumption is increasing over time.

The general form of an exponential function is:

\[ f(t) = a \cdot e^{kt} \]

where:

• \(a\) is the initial value,
• \(e\) is the base of the natural logarithm,
• \(kt\) is the exponent, and
• \(k\) is the constant growth rate.

In the exercise, the growth rate is 1.5% per year, which is written as 0.015 in decimal form. Therefore, the oil consumption rate can be modelled by the exponential function \(Rate(t) = 1.2 \cdot e^{0.015t}\). Exponential growth is critical in understanding phenomena that don't increase linearly but instead accelerate over time.

Integration is a fundamental operation in calculus that is used to find areas, volumes, central points, and many useful things, but one of its most practical applications is to determine the total accumulation of a quantity that changes continuously—such as the amount of oil consumed over time.

In the exercise, we’re given the rate of change of oil consumption as an exponential function. The process of integration allows us to find the total oil consumed over a period by accumulating the small amounts consumed at each instant within that period.

The integration of the rate function \(Rate(t) = 1.2 \cdot e^{0.015t}\) from time \(t = 0\) to time \(t\) gives us the function \(A(t) = \int Rate(t) \, dt\), which represents the total oil consumed by the country from the start of 2018 until time \(t\). This integral accumulates the instantaneous rates of consumption over time, resulting in the total consumption at any time \(t\). The solution of this integral, \(A(t) = 80 \cdot e^{0.015t} - 80\), gives us the ability to forecast total consumption at any future point, based on our continuous growth model and exponential functions.