Skip to main content
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.2.26b
Chapter 7, Problem 7.2.26b

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.
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).
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 \(C(t) = \int_0^t R(s) \, ds\).
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 \(t\).
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key 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. 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).
Recommended video:
09:29
Exponential Growth & Decay

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.
Recommended video:
04:16
Intro To Related Rates

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.
Recommended video:
04:00
Solutions to Basic Differential Equations
Related Practice
Textbook Question

Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:


b. exp(x − y) = exp(x) / exp(y)

Textbook Question

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


b. Find the total energy (in MW-yr) used by the city over four full years beginning at t=0.

Textbook Question

37–38. Caffeine After an individual drinks a beverage containing caffeine, the amount of caffeine in the bloodstream can be modeled by an exponential decay function, with a half-life that depends on several factors, including age and body weight. For the sake of simplicity, assume the caffeine in the following drinks immediately enters the bloodstream upon consumption.


An individual consumes two cups of coffee, each containing 90 mg of caffeine, two hours apart. Assume the half-life of caffeine for this individual is 5.7 hours.


b. Determine the amount of caffeine in the bloodstream 1 hour after drinking the second cup of coffee.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.


b. ln 0 = 1

Textbook Question

61–62. Points of intersection and area

b. Compute the area of the region described.


f(x) = sinh x, g(x) = tanh x; the region bounded by the graphs of f, g, and x = ln 3

1
views
Textbook Question

A slowing race Starting at the same time and place, Abe and Bob race, running at velocities u(t) = 4 / (t + 1) mi/hr and v(t) = 4e^(−t/2) mi/hr, respectively, for t ≥ 0.

b. Find and graph the position functions of both runners. Which runner can run only a finite distance in an unlimited amount of time?