Problem 10 - Active Calculus

Question

Filters at a water treatment plant become dirtier over time and thus become less effective; they are replaced every 30 days. During one 30 -day period, the rate at which pollution passes through the filters into a nearby lake (in units of particulate matter per day) is measured every 6 days and is given in the following table. The time \(t\) is measured in days since the filters were replaced. $$ \begin{array}{lcccccc} \hline \text { Day, } t & 0 & 6 & 12 & 18 & 24 & 30 \\ \hline \text { Rate of pollution in units per day, } p(t) & 7 & 8 & 10 & 13 & 18 & 35 \\ \hline \end{array} $$ Table 4.1.12: Pollution data for the water filters. a. Plot the given data on a set of axes with time on the horizontal axis and the rate of pollution on the vertical axis. b. Explain why the amount of pollution that entered the lake during this 30 -day period would be given exactly by the area bounded by \(y=p(t)\) and the \(t\) -axis on the time interval [0,30] . c. Estimate the total amount of pollution entering the lake during this 30 -day period. Carefully explain how you determined your estimate.

Step-by-Step Solution

Answer

The total estimated pollution entering the lake is 420 units.

Step 1: Plot the Data

To create a plot, use a graph with time, t, on the horizontal (x) axis and the rate of pollution, p(t), on the vertical (y) axis. Plot the given data points: (0,7), (6,8), (12,10), (18,13), (24,18), and (30,35). Draw the points and connect them with straight lines to form a piecewise linear function.

Step 2: Understand the Area Interpretation

Observe that the area under the curve of the plot represents the total amount of pollution passing through the filter because the rate of pollution, p(t), is given in units per day. The total amount of pollution is thus the integral of p(t) over the interval [0, 30].

Step 3: Trapezoidal Rule for Area Estimation

Use the trapezoidal rule to estimate the area under the curve. The trapezoidal rule approximates the area under a curve as a series of trapezoids, whose area we can add up. The trapezoidal rule formula is given by: \[ \text{Area} = \frac{1}{2} \sum_{i=1}^{n} (b_i - a_i) [f(a_i) + f(b_i)] \]Here, divide the interval [0, 30] into 5 subintervals: \[0-6, 6-12, 12-18, 18-24, 24-30\]. Each subinterval is of equal width, \Delta t = 6\ days. The area contributions from each interval are:- Interval \[0,6\]: \[ \frac{1}{2} \times 6 \times (7 + 8) = 45 \]- Interval \[6,12\]: \[ \frac{1}{2} \times 6 \times (8 + 10) = 54 \]- Interval \[12,18\]: \[ \frac{1}{2} \times 6 \times (10 + 13) = 69 \]- Interval \[18,24\]: \[ \frac{1}{2} \times 6 \times (13 + 18) = 93 \]- Interval \[24,30\]: \[ \frac{1}{2} \times 6 \times (18 + 35) = 159 \]

Step 4: Summing the Areas

Add the areas of the trapezoids together to find the total approximation:\[ \text{Total area} = 45 + 54 + 69 + 93 + 159 = 420 \]

Key Concepts

Calculus

Calculus is a branch of mathematics that helps us understand changes between variables. In this exercise, we use calculus to understand how pollution levels change over time. This is important for solving real-world problems where rates of change, like pollution rates, need to be predicted or controlled.

Calculus involves derivatives and integrals. Derivatives help us understand the rate at which something is changing, while integrals help us find the total accumulation of something, such as pollution over a time period.

Trapezoidal Rule

The trapezoidal rule is an approximate method for calculating the area under a curve. It's particularly useful when you have a set of discrete data points, as we do in this problem.

To apply the trapezoidal rule, we break the interval into smaller subintervals and approximate the area under the curve as a series of trapezoids. The area of each trapezoid is then summed to estimate the total area.

Integral Approximation

Integral approximation methods like the trapezoidal rule are essential when dealing with real-world data. Instead of continuous functions, we often have measurements at specific points.

In this problem, the rate of pollution was measured every 6 days. We use these data points to estimate the integral, which represents the total amount of pollution that entered the lake during 30 days.

The trapezoidal rule formula is:
\[ \text{Area} = \frac{1}{2} \sum_{i=1}^{n} (b_i - a_i) [f(a_i) + f(b_i)] \]
Where, \(b_i\) and \(a_i\) are the boundaries of each subinterval, and \(f(a_i)\) and \(f(b_i)\) are the function values (rates of pollution) at those points.

Data Plotting

Plotting data is a crucial step before applying mathematical solutions. By visualizing the data, we gain insights into possible trends and behaviors of the function involved.

To plot the given data:

• Take time \( t \) on the x-axis
• Rate of pollution \( p(t) \) on the y-axis
• Plot points: \( (0,7), (6,8), (12,10), (18,13), (24,18), and (30,35) \)

After plotting the data, connect the points with straight lines to form a piecewise linear function. This visual representation helps in applying the trapezoidal rule effectively.