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%/yr, starting with an initial rate of 1.2 million barrels/yr.


b. Find the function that gives the amount of oil consumed between t=0 and any future time t.

Verified step by step guidance

1. Identify the type of growth described: since the rate of oil consumption increases at a constant percentage rate (1.5% per year), this suggests exponential growth. The rate function can be modeled as an exponential function.
2. Write the rate of oil consumption function as \(R(t) = R_0 \times e^{kt}\), where \(R_0\) is the initial rate, \(k\) is the growth rate expressed as a decimal, and \(t\) is time in years. Here, \(R_0 = 1.2\) million barrels/year and \(k = 0.015\) (since 1.5% = 0.015).
3. To find the total amount of oil consumed from time \(t=0\) to any future time \(t\), integrate the rate function \(R(t)\) with respect to time over the interval \([0, t]\). This gives the consumption function \(C(t) = \int_0^t R(s) \, ds\).
4. Set up the integral explicitly: \(C(t) = \int_0^t 1.2 \times e^{0.015s} \, ds\). This integral will give the total amount of oil consumed up to time \(t\).
5. Evaluate the integral by applying the formula for integrating an exponential function: \(\int e^{as} \, ds = \frac{1}{a} e^{as} + C\). Use this to express \(C(t)\) in terms of \(t\), including the constant of integration determined by the limits of integration.

Key Concepts

Exponential Growth

Exponential growth describes a process where a quantity increases at a rate proportional to its current value. In this problem, the oil consumption rate increases by 1.5% per year, meaning the rate grows exponentially over time, modeled by a function like R(t) = R_0 * e^(kt).

Rate of Change and Integration

The rate of oil consumption is given as a function of time, and to find the total amount consumed over a period, we integrate this rate function. Integration accumulates the instantaneous rates over time, providing the total quantity consumed between two points.

Initial Conditions in Differential Equations

Initial conditions specify the starting value of a function, essential for solving differential equations uniquely. Here, the initial consumption rate at t=0 is 1.2 million barrels/year, which sets the constant in the exponential growth model for the rate.