Textbook Question
2–9. Integrals Evaluate the following integrals.
∫ (eˣ / (4eˣ + 6)) dx
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11. Integrals of Inverse, Exponential, & Logarithmic Functions
Integrals of Exponential Functions
9:57 minutesProblem 7.2.26c
Textbook 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%/yr, starting with an initial rate of 1.2 million barrels/yr.
c. How many years after 2018 will the amount of oil consumed since 2018 reach 10 million barrels?
Verified step by step guidance1
Identify the given information: the initial consumption rate at time \(t=0\) (year 2018) is \(c(0) = 1.2\) million barrels per year, and the consumption rate increases at a rate of 1.5% per year. This means the consumption rate grows exponentially.
Express the consumption rate as a function of time \(t\) (in years after 2018) using the exponential growth formula: \(c(t) = c(0) \times e^{rt}\), where \(r\) is the growth rate expressed as a decimal. Here, \(r = 0.015\).
Write the formula for the total amount of oil consumed from 2018 up to year \(t\) by integrating the consumption rate function: \(Q(t) = \int_0^t c(s) \, ds = \int_0^t 1.2 \times e^{0.015s} \, ds\).
Perform the integration to find \(Q(t)\) explicitly in terms of \(t\). This will involve integrating an exponential function.
Set the total consumption \(Q(t)\) equal to 10 million barrels and solve the resulting equation for \(t\) to find how many years after 2018 the total consumption reaches 10 million barrels.
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Video duration:
9mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Growth
Exponential growth describes a process where a quantity increases at a rate proportional to its current value, often modeled by functions like y = y_0 * e^(kt). In this problem, the oil consumption rate increases by 1.5% per year, indicating exponential growth of the consumption rate over time.
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Integration of a Rate Function
To find the total amount consumed over time, you integrate the rate function with respect to time. Since the consumption rate changes exponentially, integrating this rate from the start year to a future time gives the cumulative oil consumed up to that point.
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Solving Exponential Equations
After integrating, you get an equation involving exponential terms to represent total consumption. Solving for time requires manipulating the equation, often using logarithms, to find when the cumulative consumption reaches a specified value, such as 10 million barrels.
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