Question

mercury from an illegally dumped hazardous waste container has been leaching into the river. an environmentalist tests a number of different sites along the river for levels of mercury and records her observations in the chart.

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$$\begin{tabular}{|c|c|} \\hline \\text{sampling location} & \\text{mercury(ppb)} \\\\ \\hline 0 \\text{ km} & 300 \\\\ \\hline 1 \\text{ km downstream} & 30 \\\\ \\hline 2 \\text{ km downstream} & 3 \\\\ \\hline 3 \\text{ km downstream} & 0.3 \\\\ \\hline 4 \\text{ km downstream} & 0.03 \\\\ \\hline 5 \\text{ km downstream} & 0.003 \\\\ \\hline \\end{tabular}$$

Explanation:

Identify the variables and data points

Let \(x\) represent the distance downstream in kilometers (\(\text{km}\)).
Let \(y\) represent the mercury concentration in parts per billion (\(\text{ppb}\)).
The given data points \((x, y)\) from the table are:

• \((0, 300)\)
• \((1, 30)\)
• \((2, 3)\)
• \((3, 0.3)\)
• \((4, 0.03)\)
• \((5, 0.003)\)

Determine the relationship pattern

Observe the ratio of consecutive mercury concentration values:

$$ \frac{30}{300} = 0.1, \quad \frac{3}{30} = 0.1, \quad \frac{0.3}{3} = 0.1 $$

Since the concentration decreases by a factor of \(10\) for each \(1\text{ km}\) increase in distance, the relationship is exponential.

Formulate the exponential decay equation

The general form of an exponential decay function is:

$$ y = a \cdot b^x $$

Using the initial value at \(x = 0\):

$$ a = 300 $$

Using the decay factor determined from the ratio:

$$ b = 0.1 = 10^{-1} $$

Thus, the equation modeling the mercury concentration is:

$$ y = 300 \cdot (0.1)^x $$

Express in base 10 notation

We can also write the equation using base \(10\):

$$ y = 300 \cdot 10^{-x} $$

This equation perfectly models the concentration of mercury as a function of distance downstream.

Answer:

The relationship between the sampling location distance downstream \(x\) (in \(\text{km}\)) and the mercury concentration \(y\) (in \(\text{ppb}\)) is modeled by the exponential decay equation:

$$ y = 300 \cdot 10^{-x} $$