Question

Filling Milk Cartons A machine fills milk cartons with milk at an approximately constant rate, but backups along the assembly line cause some variation. The rates (in cases per hour) are recorded at hourly intervals during a 10 -hour period, from 8:00 A.M. to 6:00 P.M. $$ \begin{array}{cc} \hline \text { Time } & \begin{array}{c} \text { Rate } \\ (\text { cases } / \mathrm{h}) \end{array} \\ \hline 8 & 120 \\ 9 & 110 \\ 10 & 115 \\ 11 & 115 \\ 12 & 119 \\ 1 & 120 \\ 2 & 120 \\ 3 & 115 \\ 4 & 112 \\ 5 & 110 \\ 6 & 121 \\ \hline \end{array} $$ Use the Trapezoidal Rule with \(n=10\) to determine approximately how many cases of milk were filled by the machine over the 10 -hour period. 1156.5

Short answer

The approximate number of milk cases filled by the machine over the 10-hour period is 1156.5.

Step-by-step solution

Understanding the Trapezoidal Rule

For a function \(f(x)\) defined on a domain \([a, b]\) with \(n\) subintervals, the trapezoidal rule estimates the area under the curve by using formulas which sum up areas of trapezoids. The formula is: \[ T_n = \frac{b - a}{2n}\left[f(a) + 2f(a+h)+2f(a+2h)+...+2f(a+(n-1)h) + f(b)\right] \] where \(h = (b - a)/n\) is the width of each subinterval.

Application of Trapezoidal Rule

In this case, \(a=8\), \(b=18\) (as 6PM is the 18th hour in a 24-hour format), \(n=10\), and \(f(x)\) represents the rate of cases filled for each hour. The 'x' values are the hours at hourly intervals and the 'y' values are the respective fill rates. Therefore, the formula becomes: \[ T_{10} = \frac{18 - 8}{2*10}\left[120+2(110+115+115+119+120+120+115+112+110)+121\right] \]

Calculation

Now, you can input the numbers into the formula. After doing the calculation, the approximate number of milk cases filled by the machine will be found.

Key concepts

Integration Approximation

When dealing with real-world situations, such as a machine filling milk cartons at varying rates, it's often necessary to estimate quantities that are difficult to measure directly. In calculus, the process of measuring a quantity like the total amount of milk filled over time can be interpreted as finding the area under a curve of a rate function on a graph. However, with actual data, we don't always have a smooth curve to integrate. Here, integration approximation techniques like the Trapezoidal Rule come in handy.

Integration approximation is essentially a mathematical strategy to estimate the definite integral, which is the total accumulation of a quantity over an interval. The Trapezoidal Rule does this by breaking down the domain (the time from 8 A.M. to 6 P.M. in our exercise) into smaller intervals and approximating the area under the curve using trapezoids rather than the exact curve. This results in a more manageable calculation that provides an estimate that's often sufficiently accurate for practical purposes.

Applying the Trapezoidal Rule allows us to tackle problems where data is available only in discrete form, like the hourly recorded rates of milk filling, rather than as a continuous function.

Numerical Methods in Calculus

Numerical methods in calculus are a set of powerful tools used to find approximate solutions to problems involving continuous functions. They're especially useful when finding exact solutions is too complex, impossible, or when dealing with empirical data. Among these methods, the Trapezoidal Rule is a popular technique because it balances simplicity with precision.

Numerical methods also include other algorithms like Simpson's Rule, Euler's Method, and Newton's Method. Each of these utilizes different strategies for approximation. Regardless of the method, the key idea is to replace the complex or unknown function with a simpler one that's easier to analyze. Using numerical methods is particularly effective in engineering, physics, and economics, where the variables in play make it difficult to model situations with simple functions.

In the context of the exercise, the Trapezoidal Rule is chosen over other numerical methods due to its direct application to the available data—hourly rates. The formula divides the task into manageable parts, leading to an overall approximation of the total filled cases of milk without needing a precise mathematical description of how the rate changes from hour to hour.

Rate of Change

The rate of change is a foundational concept in calculus that describes how a quantity changes with respect to another. In our exercise, the rate of change pertains to how the number of filled milk cartons changes over time. Graphically, this rate of change is represented by the slope of the line on a graph where the x-axis is time and the y-axis is the quantity of cases filled per hour.

When looking at the problem of filling milk cartons, each interval's rate of change tells us about the productivity during that hour. The machine's fill rate fluctuates, and these variances impact the total output. By summing up these rates, we account for both the steady and fluctuating intervals of production.

The Trapezoidal Rule allows us to take these variable rates into account by connecting them through straight-line segments, forming trapezoids. The area of each trapezoid is a reflection of the total products produced in that time period. In essence, by using the Trapezoidal Rule to sum these areas, we're summing up the instantaneous rates of change over discrete intervals, thus quantifying the overall change in quantity - in this case, the total number of milk cases filled over the 10-hour period.