Question

Multiple Choice Pollution is being removed from a lake at a rate modeled by the function \(y=20 e^{-0.5 t}\) tons/yr, where \(t\) is the number of years since \(1995 .\) Estimate the amount of pollution removed from the lake between 1995 and \(2005 .\) Round your answer to the nearest ton. \(A\) (A) 40 (B) 47 (C) 56 (D) 61 (E) 71

Short answer

To provide the short answer, calculate the total pollution removed which is obtained in Step 3, and look for the closest matching option.

Step-by-step solution

Setting up the integral

First, we note that the number of years since 1995 is given by \(t\), so that for the years 1995 to 2005, we have \(t=0\) to \(t=10\). Because the function represents the rate at which pollution is being removed from the lake, the total amount of pollution removed over the period is the integral of the function over this interval. So, we need to evaluate the definite integral \(\int_{0}^{10} 20 e^{-0.5 t} dt\).

Evaluating the integral

The integral can be evaluated using the fundamental theorem of Calculus. The antiderivative of \(e^{-0.5 t}\) is \(-2e^{-0.5 t}\). So, the integral \(\int 20e^{-0.5 t} dt = -40e^{-0.5t}\). We now need to evaluate this antiderivative at the limits of integration 0 and 10. This gives \([-40 e^{-0.5 \cdot 10}]-[-40 e^{-0.5 \cdot 0}]\).

Calculating the total

Our next step is to compute \([-40 e^{-0.5 \cdot 10}]-[-40 e^{-0.5 \cdot 0}]\). Simplifying these expressions gives us \[-40 e^{-5} + 40\]. After calculating the value of this expression, round your answer to the nearest integer.

Final answer

The computed total will represent the estimated tons of pollution removed from the lake between the years 1995 and 2005. Match your answer with the given options.

Key concepts

Integral Calculus

Integral calculus, a fundamental branch of mathematical analysis, deals with the accumulation of quantities, such as areas under curves, total accumulated change, or in this context, the total amount of a substance removed over time.

Imagine pouring water into a bathtub; integral calculus helps us figure out not just the rate at which water is entering the tub but also how much water has been added over a period of time. Similarly, in our pollution removal problem, although we are given a rate of removal, we seek to calculate the total volume of pollution removed from the lake.

To find this total, we use a definite integral, which enables us to determine the accumulated quantity between two points in time. In essence, the process of integration 'adds up' all the small quantities removed over each instant within our interval from 1995 to 2005, providing us with a complete picture of the environmental cleanup effort over these ten years.

Exponential Decay Model

The exponential decay model is often employed to describe processes where quantities decrease at a rate proportional to their current value. This type of behavior is common in radioactive decay, population decline, and, as seen in this problem, environmental pollution reduction.

In our scenario, the model given by the function ( y=20e^{-0.5t} ) represents how pollution is removed. The base of the exponential function (e) is the mathematical constant approximately equal to 2.71828, and it's widely used due to its unique properties in growth and decay processes. The rate of decay here is proportional, indicated by the negative exponent (-0.5t), which reflects a diminishing removal rate over time.

Understanding this model allows us to predict the future condition of the lake and provides invaluable insight into the efficiency of clean-up operations as time progresses.

Environmental Application of Calculus

Applying calculus to environmental science allows us to create models that mimic real-world phenomena. Models assist in the prediction, evaluation, and strategizing of environmental efforts like pollution removal, resource consumption, and population dynamics.

In practice, by quantifying the rate at which pollutants are removed from an ecosystem, environmentalists can gauge the progress of conservation measures, optimize resource allocation, and even project recovery timelines for affected areas. These applications underscore how calculus isn't just abstract mathematics but a vital tool for making informed decisions that can impact the health of our planet.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the seemingly separate subjects of differentiation and integration. It tells us that if you take the integral of a function (which gives you accumulated change) and then differentiate it (which gives you instant rate of change), you will get the original function back.

Moreover, this theorem allows us to evaluate definite integrals in a straightforward manner. By finding an antiderivative of the function we are integrating and evaluating it at the upper and lower bounds, we obtain the total accumulation over the interval. In relation to the pollution problem, this theorem simplifies the process of calculating the total amount of pollution removed over a set period of time by simply evaluating the antiderivative we find at both the start and end points of our defined interval (1995 to 2005), which is essentially what steps 2 and 3 in the given solution are demonstrating.