91Ó°ÊÓ

Cyanide is used in solution to isolate gold in a mine. \(^{49}\) This may result in contaminated groundwater near the mine, requiring the poison be removed, as in the following table, where \(t\) is in years since 2012. (a) Find an exponential model for \(c(t),\) the concentration, in parts per million, of cyanide in the groundwater. (b) Use the model in part (a) to find the number of years it takes for the cyanide concentration to fall to 10 ppm. (c) The filtering process removing the cyanide is sped up so that the new model is \(D(t)=c(2 t) .\) Find \(D(t).\) (d) If the cyanide removal was started three years earlier, but run at the speed of part (a), find a new model, \(E(t).\) $$\begin{array}{c|c|c|c} \hline t \text { (years) } & 0 & 1 & 2 \\ \hline c(t)(\mathrm{ppm}) & 25.0 & 21.8 & 19.01 \\ \hline \end{array}$$

Step by step solution

01

Determine the Exponential Model

To find the exponential model, we assume that the concentration function is of the form \( c(t) = c_0 e^{kt} \), where \( c_0 \) is the initial concentration and \( k \) is the rate constant. We know that \( c(0) = 25 \) ppm, \( c(1) = 21.8 \) ppm, and \( c(2) = 19.01 \) ppm. Using these, we can set up two equations: 1. \( 25 e^{k} = 21.8 \) 2. \( 21.8 e^{k} = 19.01 \) From equation 1, we solve for \( e^{k} \) which is \( \frac{21.8}{25} \). Now, substitute \( e^{k} \) into equation 2: \\[ 21.8 \left( \frac{21.8}{25} \right) = 19.01 \]. \ Simplifying, the value of \( k \) can be found by solving: \[ k = \ln \left( \frac{21.8}{25} \right) \approx -0.142 \] . Thus, the exponential model is: \( c(t) = 25e^{-0.142t} \).

02

Find Years to Reach 10 ppm

Using the model from Step 1, we want to find \( t \) such that \( c(t) = 10 \) ppm. Substitute 10 for \( c(t) \): \[ 10 = 25 e^{-0.142 t} \] Divide both sides by 25: \[ \frac{10}{25} = e^{-0.142 t} \] \[ 0.4 = e^{-0.142 t} \] Take the natural logarithm of both sides: \[ \ln(0.4) = -0.142 t \] Solve for \( t \): \[ t \approx \frac{\ln(0.4)}{-0.142} \approx 6.45 \]. So, it takes approximately 6.45 years for the concentration to reach 10 ppm.

03

Find the New Model D(t)

The new model \( D(t) \) is given by \( D(t) = c(2t) \). Using the model \( c(t) = 25 e^{-0.142t} \), substitute \( 2t \) for \( t \):\[ D(t) = 25 e^{-0.142(2t)} = 25 e^{-0.284t} \].

04

Find Model E(t) with Earlier Start

If the process started three years earlier, the concentration function would adjust accordingly. For a model \( E(t) \) that started three years earlier, we need to shift the function \( c(t) = 25 e^{-0.142t} \) to the right by 3 years:\[ E(t) = c(t+3) = 25 e^{-0.142(t+3)} \] Simplify to get the new model: \[ E(t) = 25 e^{-0.142t - 0.426} \] This accounts for the earlier start by effectively increasing the exponent by \(-0.426\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cyanide Concentration

Cyanide concentration refers to the amount of cyanide present in groundwater, measured in parts per million (ppm). In the context of mining operations, particularly gold extraction, cyanide is used in solution for its effectiveness in separating gold from ore. However, the downside is potential contamination in nearby groundwater, requiring measures to monitor and reduce its levels. By understanding the concentration levels over time, we can actively manage and mitigate the adverse impacts on environmental health. The initial concentration of cyanide can serve as a baseline for creating mathematical models that predict future levels of contamination through the use of exponential equations.

Groundwater Contamination

Groundwater contamination occurs when hazardous substances, such as cyanide from mining activities, seep into the groundwater supply. This can pose serious environmental and health risks. The contaminants can spread over time, impacting water quality for surrounding communities and ecosystems. Monitoring and modeling are essential to comprehend how such pollutants dilute, disperse, or decay over time in the subsurface. By applying a mathematical model to the problem, one can estimate the rate of decrease in contaminant concentration and make informed decisions about remediation strategies and timelines. These predictive models aid in determining the effectiveness of current filtration processes to safeguard water sources.

Rate Constant

The rate constant, denoted as \( k \), is a crucial component in exponential decay equations. It quantifies the speed at which the cyanide concentration decreases over time. In modeling the exponentially decaying cyanide concentration, the rate constant is derived from observed data points, such as those provided at specific yearly intervals in the exercise. To compute \( k \), one takes the natural logarithm of the ratio of concentrations from consecutive time points. A negative value of \( k \) indicates the decay nature of the process, showcasing how swiftly the concentration reduces with passing time. An accurate calculation of \( k \) allows for reliable predictions and adjustments in treatment methods.

Exponential Decay

Exponential decay describes a process where the quantity decreases at a rate proportional to its current value. In this case, it applies to the reduction in cyanide concentration in groundwater over time. The mathematical model utilized is of the form \( c(t) = c_0 e^{kt} \), where \( c_0 \) is the initial concentration and \( k \) is the rate constant. This model predicts how quickly the cyanide levels fall, providing essential information for environmental management. Exponential decay models are powerful because they reflect the natural logarithmic decrease of substances over time, offering clarity and precision in projections. Understanding this concept is vital for efficiently planning purification efforts and evaluating contamination levels periodically.