Problem 6 - Applied Calculus

Question

Pollution is removed from a lake on day \(t\) at a rate of \(f(t) \mathrm{kg} /\) day. (a) Explain the meaning of the statement \(f(12)=500\). (b) If \(\int_{5}^{15} f(t) d t=4000,\) give the units of the \(5,\) the \(15,\) and the 4000. (c) Give the meaning of \(\int_{5}^{15} f(t) d t=4000\).

Step-by-Step Solution

Answer

(a) f(12)=500 means pollution is removed at 500 kg/day on day 12. (b) Units: days for 5 and 15, kg for 4000. (c) 4000 kg of pollution removed from day 5 to 15.

Step 1: Interpret the Statement f(12)=500

The statement \( f(12) = 500 \) means that at day 12, the rate at which pollution is being removed from the lake is 500 kg/day. In this context, \( f(t) \) is a function representing the rate of pollution removal per day at day \( t \).

Step 2: Identify Units for the Integral and Limits

In the integral \( \int_{5}^{15} f(t) \, dt = 4000 \), the limits 5 and 15 represent time in days. Specifically, day 5 and day 15 respectively. The value 4000 represents the total amount of pollution removed, which is measured in kilograms.

Step 3: Explain the Meaning of the Integral

The statement \( \int_{5}^{15} f(t) \, dt = 4000 \) means that over the ten-day period from day 5 to day 15, a total of 4000 kg of pollution was removed from the lake. The integral sums up the rate of removal over this period to find the total amount removed.

Key Concepts

Rate of Change

The rate of change is a concept used to describe how a quantity changes with respect to another variable. In the context of the exercise, the rate of change is the speed at which pollution is removed from the lake per day. This is denoted by the function \( f(t) \), where \( t \) stands for a particular day. For example, the statement \( f(12) = 500 \) kg/day tells us that precisely on day 12, 500 kg of pollution is being removed. The rate of change helps us understand how quickly or slowly changes occur in a system, and it's often measured in units like kg/day, revealing the dynamics of environmental processes.

Rate of change formulas, like derivatives, help us compute these values over an infinitesimally small time frame, giving an instantaneous view of the system's behavior. This understanding is crucial when planning environmental restoration or understanding pollutant dynamics.

Units of Measure

Units of measure are fundamental in science and engineering as they give meaning to numerical results. In definite integrals involving pollution removal rates, several units of measure are at play. In the exercise, \( \int_{5}^{15} f(t) dt = 4000 \) outlines these units:

• The numbers 5 and 15 signal specific days, so they represent time in days.
• The number 4000, being the result of the integral, signifies the total amount of pollution removed and is measured in kilograms.
• The function \( f(t) \), defining the rate, uses kg/day as its unit to express the rate of pollution removal over time.

Understanding these units is vital, as they ensure the consistency and correctness of calculations. Without proper units, the numerical values lose their significance, leading to potential misunderstandings in data interpretation.

Definite Integral

The definite integral is a core concept in calculus that allows the summation of an infinite number of infinitesimally small data points to find one comprehensive value. It integrates a function over an interval and can give concrete real-world insights. In the exercise, the definite integral \( \int_{5}^{15} f(t) dt = 4000 \) tells us about the pollution dynamics between days 5 and 15.

The function \( f(t) \) represents the removal rate, and integrating from day 5 to day 15 helps us determine the total amount of pollution removed during these days. This total, 4000 kg, is not merely an approximation but an exact result signifying the accumulation of pollution removed over this period.

The interpretation of definite integrals is useful in contexts like:

• Calculating total change over time, like population growth or decay.
• Determining displacement when given a velocity-time function.
• Calculating area under a curve in various scientific fields.

This makes the definite integral an invaluable tool in both theoretical and applied mathematics.