Problem 37
Question
An oil refinery produces oil at a variable rate given by $$Q^{\prime}(t)=\left\\{\begin{array}{ll}800 & \text { if } 0 \leq t<30 \\\2600-60 t & \text { if } 30 \leq t<40 \\\200 & \text { if } t \geq 40,\end{array}\right.$$ where \(t\) is measured in days and \(Q\) is measured in barrels. a. How many barrels are produced in the first 35 days? b. How many barrels are produced in the first 50 days? c. Without using integration, determine the number of barrels produced over the interval [60,80].
Step-by-Step Solution
Verified Answer
By following the solution, we found that:
$$ Q(35) = Q_{0-30} + Q_{30-35} $$
Calculate the values:
$$ Q_{0-30} = 800\times30 = 24000 $$
$$ Q_{30-35} = (2600\times35 - 30\times35^2) - (2600\times30 - 30\times30^2) = 5425 $$
Therefore:
$$ Q(35) = 24000 + 5425 = 29425 $$
The refinery produced 29,425 barrels of oil in the first 35 days.
b) How many barrels of oil were produced in the first 50 days?
We have calculated the barrels produced in the first 35 days, so now we need to find the barrels produced between 40 and 50 days:
$$ Q_{40-50} = 200\times10 = 2000 $$
Calculate the total barrels produced in the first 50 days:
$$ Q(50) = Q(35) + Q_{40-50} = 29425 + 2000 = 31425 $$
The refinery produced 31,425 barrels of oil in the first 50 days.
c) How many barrels of oil were produced between days 60 and 80 without using integration?
Since the rate is a constant 200 barrels/day for 20 days, we can find the total barrels produced between 60 and 80 days:
$$ Q_{60-80} = 200 \times 20 = 4000 $$
The refinery produced 4,000 barrels of oil between days 60 and 80 without using integration.
1Step 1: Part a - Barrels produced in the first 35 days
To find the number of barrels produced in the first 35 days, calculate the total barrels produced during the individual sub-intervals (0-30 days and 30-35 days).
For the first 30 days, the rate is constant at 800 barrels/day, so the total barrels produced will be:
$$ Q_{0-30} = 800\times30 $$
For the next 5 days (30-35), the rate is given by the function:
$$ Q'(t) = 2600 - 60t $$
Integrate this function with respect to \(t\) in the interval [30,35]:
$$ Q_{30-35} = \int_{30}^{35} (2600 - 60t) dt $$
Calculate the definite integral:
$$ Q_{30-35} = [2600t - 30t^2]_{30}^{35} $$
$$ Q_{30-35} = (2600\times35 - 30\times35^2) - (2600\times30 - 30\times30^2) $$
Now calculate Q (35):
$$ Q(35) = Q_{0-30} + Q_{30-35} $$
2Step 2: Part b - Barrels produced in the first 50 days
Now, find the barrels produced in the first 50 days. The time intervals and their respective rates are (0-30 days, 800 barrels/day), (30-40 days, 2600 - 60t barrels/day), and (40-50 days, 200 barrels/day).
We have already calculated the total barrels produced in the first 35 days, \(Q(35)\).
For the next 5 days (40-45), the rate is constant at 200 barrels/day, so the total barrels produced will be:
$$ Q_{40-50} = 200\times10 $$
Now, calculate the total barrels produced in the first 50 days:
$$ Q(50) = Q(35) + Q_{40-50} $$
3Step 3: Part c - Barrels produced between 60 and 80 days without integration
To find the number of barrels produced in the interval [60,80] without using integration, notice that the rate is constant during this interval as it is mentioned in the production rate function:
$$ Q'(t)=200\ \text{if}\ t\geq40 $$
In this [60,80] interval, the rate is a constant 200 barrels/day for 20 days:
$$ Q_{60-80} = 200 \times 20 $$
Hence, the number of barrels produced over the interval [60,80] is:
$$ Q_{60-80} $$
Key Concepts
Definite IntegralPiecewise FunctionRate of ChangeCalculating Total Production
Definite Integral
The concept of a definite integral is fundamental when calculating the total quantity of something given a rate of change. In this exercise, the definite integral is used to determine the total amount of oil produced over specific intervals of time, given a variable rate of production. The rate of oil production is provided as a function of time, and by integrating this function over given intervals, we obtain the total oil produced during those times.
For instance, to find the total production between two points in time—like from day 30 to day 35—we integrate the rate function from day 30 to 35. In mathematical terms, if we have a production rate function \( Q^{\prime}(t) \), then the total production over an interval \([a, b]\) is given by the integral, \( \int_{a}^{b} Q^{\prime}(t) \, dt \). This integral essentially sums up small amounts of production within each infinitesimally small sub-interval from \(a\) to \(b\), giving the total production.
For instance, to find the total production between two points in time—like from day 30 to day 35—we integrate the rate function from day 30 to 35. In mathematical terms, if we have a production rate function \( Q^{\prime}(t) \), then the total production over an interval \([a, b]\) is given by the integral, \( \int_{a}^{b} Q^{\prime}(t) \, dt \). This integral essentially sums up small amounts of production within each infinitesimally small sub-interval from \(a\) to \(b\), giving the total production.
Piecewise Function
A piecewise function is a function that has different expressions for different parts of its domain. In this exercise, the rate of oil production is defined as a piecewise function where the rate changes in specific intervals of days. This can be likened to a factory that shifts gears based on demand or operational capacity.
The piecewise function here is written to cover three distinct time frames: for \(0 \leq t < 30\), for \(30 \leq t < 40\), and for \(t \geq 40\). Each segment of the function is different: it is a constant rate for the first and last intervals and a linearly decreasing function in the middle segment.
Understanding and interpreting piecewise functions is crucial, as it allows one to apply different calculations directly corresponding to each piece or interval where the function behaves distinctly.
The piecewise function here is written to cover three distinct time frames: for \(0 \leq t < 30\), for \(30 \leq t < 40\), and for \(t \geq 40\). Each segment of the function is different: it is a constant rate for the first and last intervals and a linearly decreasing function in the middle segment.
Understanding and interpreting piecewise functions is crucial, as it allows one to apply different calculations directly corresponding to each piece or interval where the function behaves distinctly.
Rate of Change
The term 'rate of change' refers to how a quantity is varying with respect to time. In the context of this exercise, it describes how the oil production rate—expressively given as barrels per day—changes with time. It's essential to interpret graphs and functions that represent rates of change to understand their implications in real-world scenarios.
The rate, represented mathematically as \( Q^{\prime}(t) \), can be constant or variable. Here, we observe variable rates during the interval \(30 \leq t < 40\), indicating changes in production strategy or capacity. This type of analysis reflects common real-world situations, where operations are adjusted based on various factors, potentially altering production rates.
The rate, represented mathematically as \( Q^{\prime}(t) \), can be constant or variable. Here, we observe variable rates during the interval \(30 \leq t < 40\), indicating changes in production strategy or capacity. This type of analysis reflects common real-world situations, where operations are adjusted based on various factors, potentially altering production rates.
Calculating Total Production
Calculating total production involves determining the total output over a specified period. When given a rate of production, this is typically done through integration if the rate function varies, or by simple multiplication if the rate is constant.
In this exercise, two aspects of total production calculations are highlighted: using integral calculus to manage variable rates and simple arithmetic for constant rates.
In this exercise, two aspects of total production calculations are highlighted: using integral calculus to manage variable rates and simple arithmetic for constant rates.
- For time intervals where the production rate is constant, the total production is found by multiplying the rate by the number of days within that interval.
- For variable rates, integration is employed. This was demonstrated for finding production between days 30 and 35, where the rate function \(2600 - 60t\) was integrated.
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