Question

Oil consumption Starting in \(2018(t=0),\) the rate at which oil is consumed by a small country increases at a rate of \(1.5 \% / \mathrm{yr}\), start. ing with an initial rate of 1.2 million barrels/yr. a. How much oil is consumed over the course of the year 2018 (between \(t=0\) and \(t=1\) )? b. Find the function that gives the amount of oil consumed between \(t=0\) and any future time \(t\) c. How many years after 2018 will the amount of oil consumed since 2018 reach 10 million barrels?

Short answer

Answer: Approximately 78.91 years.

Step-by-step solution

Calculate oil consumption over 2018

To find the oil consumed over 2018, we need to use the given rate of 1.2 million barrels/yr as the consumption increases at a rate of 1.5%/yr. Since the year is just one year long, the consumption rate remains constant at 1.2 million barrels/yr. Therefore, the amount of oil consumed over 2018 would be 1.2 million barrels.

Calculate annual consumption as a function of time

We are given that the rate of oil consumption increases by 1.5% per year.Using the continuous growth formula: \(Rate(t)=Rate_0e^{rt}\), where \(Rate_0\) is the initial rate, \(t\) is the time in years, \(r\) is the percentage increase per year, and \(Rate(t)\) is the rate of oil consumption at a certain time. \(Rate(t)=1.2e^{0.015t}\)

step 3: Find the total oil consumed as a function of time

First, let's convert the rate of oil consumption into the amount of oil consumed by integrating over time. Since the rate of oil consumption represents the derivative of the amount of oil consumed with respect to time, we will integrate \(Rate(t)\) with respect to \(t\) to find the amount: \(A(t) = \int_{0}^{t} Rate(t) dt = \int_{0}^{t} 1.2e^{0.015t} dt\) On integrating, we get: \(A(t) = 1.2 \cdot \frac{e^{0.015t}}{0.015}\Big|_0^t = 80e^{0.015t} - 80\) This function represents the total oil consumed from \(t=0\) (2018) to any given future time \(t\).

Find the time when oil consumption reaches 10 million barrels

We need to find the value of \(t\) when the total oil consumed, \(A(t)\), reaches 10 million barrels: \(10,000,000 = 80e^{0.015t} - 80\) Solve for \(t\): \(10,000,080 = 80e^{0.015t}\) \(\frac{10,000,080}{80} = e^{0.015t}\) Now, take the natural logarithm of both sides: \(\ln\left(\frac{10,000,080}{80}\right) = 0.015t\) Now, divide by 0.015: \(t = \frac{\ln\left(\frac{10,000,080}{80}\right)}{0.015}\) Using a calculator, the result is approximately: \(t \approx 78.91\) Therefore, the amount of oil consumed since 2018 will reach 10 million barrels approximately 78.91 years after 2018.

Key concepts

Continuous Growth Model

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

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 in Calculus

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.