/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 The rate of oil production by Pe... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 63

The rate of oil production by Pemex, Mexico's national oil company, can be approximated by \(q(t)=-8 t^{2}+70 t+1,000\) million barrels per year $$ (0 \leq t \leq 9) $$ where \(t\) is time in years since the start of \(2000 .{ }^{8}\) During that time, the price of oil was approximately \(^{9}\) \(p(t)=25 e^{0.1 t}\) dollars per barrel. Obtain an expression for Pemex's total oil revenue \(R(x)\) since the start of 2000 to the start of year \(x\) as a function of \(x\). (Do not simplify the answer.) HINT [Rate of revenue \(=p(t) q(t)\).]

Short Answer

Expert verified
The total oil revenue function, R(x), for Pemex since the start of 2000 to the start of year x as a function of x is given by: \( R(x) = \int_0^x (25e^{0.1t})(-8t^2 + 70t + 1000) dt \)

Step by step solution

01

Write down the given functions

We are given the following functions for oil production rate, q(t), and oil price, p(t): \( q(t) = -8t^2 + 70t + 1000 \) million barrels per year \( p(t) = 25e^{0.1t} \) dollars per barrel
02

Find the rate of revenue

The rate of revenue can be calculated by multiplying the oil production rate with the oil price: Rate of revenue = p(t)q(t) \( r(t) = (25e^{0.1t})\times(-8t^2 + 70t + 1000) \)
03

Integrate the rate of revenue function

To find the accumulated revenue, R(x), we need to integrate the rate of revenue function, r(t), with respect to time t from 0 to x: \( R(x) = \int_0^x r(t) dt \) \( R(x) = \int_0^x (25e^{0.1t})(-8t^2 + 70t + 1000) dt \)
04

Present the final result

Pemex's total oil revenue R(x) since the start of 2000 to the start of year x as a function of x is: \( R(x) = \int_0^x (25e^{0.1t})(-8t^2 + 70t + 1000) dt \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration
Integration is a fundamental concept in calculus, essential for calculating areas under curves and solving problems involving accumulation over time. In the context of revenue calculations, integration helps us find the total accumulated revenue from a varying rate of revenue.
For example, when faced with a function like the rate of revenue, denoted as \( r(t) = (25e^{0.1t})\times(-8t^2 + 70t + 1000) \), integration allows us to sum up the continuous stream of revenue over a specific period.
  • To perform integration, you need an integrand (the function to integrate) and limits of integration (the interval over which to integrate).
  • In this exercise, our integrand is the rate of revenue function \( r(t) \) and the limits are from 0 to \( x \) years.
The notation \( \int_0^x \) indicates that we integrate from the start of year 0 to the start of year \( x \). This process gives us the total revenue function \( R(x) \). Integration naturally accumulates area under the rate curve, representing the revenue collected.
Revenue Calculations
Revenue calculations often involve determining how much money a company makes over time. In the given problem, revenue is calculated as the product of the quantity of oil produced and the price of oil per barrel.
The rate of revenue is expressed as \( r(t) = p(t)q(t) \), where \( p(t) \) is the price function \( 25e^{0.1t} \) and \( q(t) \) is the production rate function \( -8t^2 + 70t + 1000 \).
  • The product of these functions results in a new function that represents the dollar amount earned per year from oil production.
  • Integration of this rate function over time allows us to calculate the total revenue collected from the start of 2000 to year \( x \).
This integrated value, \( R(x) \), tells us how much total revenue Pemex garners over a specified period given changes in production rates and oil pricing.Understanding revenue in this consolidated numerical form helps businesses make informed financial decisions.
Exponential Functions
Exponential functions are crucial in modeling scenarios where growth or decay rates change by a constant percentage. An exponential function in the form \( p(t) = 25e^{0.1t} \) describes how the price of oil changes over time.
In this specific case, the base \( e \) is Euler's number, approximately equal to 2.71828, and it signifies continuous growth. The exponent \( 0.1t \) implies a consistent 10% annual growth rate in oil prices.
  • Given the exponential growth of prices, revenues are also impacted significantly over longer time frames due to compounding effects.
  • These functions are quite sensitive to changes in the exponent, which can lead to notable increases in the output value as time progresses.
Exponential functions model real-world phenomena, such as economic growth and decay, where the rate of change is proportional to the current state. By using exponential functions in revenue calculations, one can predict financial outcomes under varying economic conditions.

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Most popular questions from this chapter

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