Question

The cost \(C\) (in millions of dollars) for the federal government to seize \(p \%\) of a type of illegal drug as it enters the country is modeled by \(C=528 p /(100-p), \quad 0 \leq p<100\) (a) Find the costs of seizing \(25 \%, 50 \%\), and \(75 \%\). (b) Find the limit of \(C\) as \(p \rightarrow 100^{-}\). Interpret the limit in the context of the problem. Use a graphing utility to verify your result.

Short answer

The costs of seizing 25%, 50%, and 75% of the drug are $176M, $528M, and $1584M respectively. As the percentage \(p\) of drugs seized approaches 100, the cost \(C\) tends towards infinity. This affirms that aiming for a 100% seizure rate, while potentially advantageous in terms of drug control would be economically unfeasible.

Step-by-step solution

Calculating the costs

The cost function given is \(C=528 p/(100-p)\). To find the costs of seizing \(25\%, 50\%\), and \(75\%\) of the drug, plug these values into the function:When \(p=25\), \(C=528*25/(100-25) = \$176M\)When \(p=50\), \(C=528*50/(100-50) = \$528M\)When \(p=75\), \(C=528*75/(100-75) = \$1584M\)

Calculating the limit

To find the limit of \(C\) as \(p\) approaches \(100^{-}\), we should remember that such a limit represents the cost of seizing almost all (just shy of 100%) of the drug. Mathematically, this is represented as \(\lim_{p \to 100^-}C\). Given the cost function \(C=528p/(100-p)\), as \(p\) approaches 100 from the left, the denominator \(100 - p\) approaches 0. As a result, the cost function tends towards infinity (as division by very small amounts results in very large numbers), formally, \(\lim_{p \to 100^-}C = \infty\)

Interpretation of the result

The result implies that as the percentage of the drug seized approaches 100%, the cost for the federal government to do so would become prohibitively large, tending towards infinity. This means it is not economically feasible to aim for a 100% seizure rate.

Verification using a graphing utility

Using a graphing utility to plot the function \(C=528p/(100-p)\) for \(0 \leq p < 100\), would show that the graph approaches infinity as \(p\) gets very close but less than 100. This reaffirms the result obtained mathematically in step 2.

Key concepts

Cost Function

A cost function in economics describes the total cost of producing a certain quantity of a good or service. Here, the function given is specifically for calculating the cost of seizing illegal drugs, represented by \[C = \frac{528p}{100-p}\]where \(C\) is the cost in millions of dollars, and \(p\) is the percentage of the illegal drug seized.

To use this function:

• Substitute the value of \( p \) into the function.
• Calculate the denominator (\(100 - p\)).
• Divide the product of 528 and \(p\) by the denominator to find \(C\).

For example, when \(p = 25\), substituting this into the formula gives \(C = 528 \times 25 / (100 - 25) = 176\) million dollars. This allows policymakers to estimate the financial burden of implementing such seizure efforts. Understanding the cost function is crucial for decision-making and resource allocation.

Limits

In calculus, a limit describes the value that a function approaches as the input approaches a certain value. In this exercise, we explore the limit of the cost function \[\lim_{p \to 100^-} C\]This limit addresses what happens to the cost \(C\) as the seizing percentage \(p\) approaches 100 from the left. As \(p\) increases towards 100, the denominator \(100 - p\) becomes very small.

Mathematically, when dividing by a number close to zero, the result is a very large number, suggesting that the cost \(C\) tends to infinity. This implies that seizing nearly 100% of the drugs would result in prohibitively high costs, signaling a point of diminishing returns. Limits help us understand behavior in these boundary conditions, especially when direct calculation isn't feasible.

Graph Interpretation

Graphing is a powerful tool to visualize mathematical functions and their behaviors. For the function \[ C = \frac{528p}{100-p} \]we can use a graph to interpret the effects of changing \(p\) on the cost \(C\). As you graph this function for values of \(p\) from 0 to just under 100, you'll observe that:

• The graph steeply rises as \(p\) nears 100.
• The curve appears to approach a vertical asymptote at \(p = 100\).

This graphical representation reaffirms what the calculus limit analysis reveals: as \(p\) approaches 100, \(C\) heads towards infinity.

Interpreting graphs allows you to grasp complex relationships quickly and spot possible trends, costs, and limits at a glance. This tool is invaluable in economics to assess the feasibility and efficiency of different strategies visually.