Problem 67 - College Algebra and Calculus: An Applied Approach

Question

MAKE A DECISION: SEIZURE OF ILLEGAL DRUGS The cost \(C\) (in millions of dollars) for the federal government to seize \(p\) percent of an illegal drug as it enters the country is \(C=\frac{528 p}{100-p}, \quad 0 \leq p<100\) (a) Find the cost of seizing \(25 \%\) of the drug. (b) Find the cost of seizing \(50 \%\) of the drug. (c) Find the cost of seizing \(75 \%\) of the drug. (d) According to this model, would it be possible to seize \(100 \%\) of the drug? Explain.

Step-by-Step Solution

Answer

a) $176 million, b) $528 million, c) $1584 million, d) No, it would not be possible as the cost becomes undefined.

Step 1: Calculate the cost for 25%

To find out the cost for seizing 25% of the illegal drug, the specific value of \(p\), which is 25, needs to be plugged into the given formula that is \(C=\frac{528 p}{100-p}\). Calculating the expression results in \(C=\frac{528*25}{100-25} = \$176\) million.

Step 2: Calculate the cost for 50%

Similarly, to find out the cost for seizing 50% of the illegal drug, the value of \(p\), which is 50, needs to be plugged into the given formula. In doing so, the expression \(C=\frac{528*50}{100-50}\) results in C=\$528 million.

Step 3: Calculate the cost for 75%

To find out the cost for seizing 75% of the illegal drug, the value of \(p\), which is 75, needs to be plugged into the given formula. In doing so, the expression \(C=\frac{528*75}{100-75}\) results in C=\$1584 million.

Step 4: Analyze the cost for 100%

According to the given model, in order to seize 100% of the illicit drug, the value of \(p\) which is 100, would be plugged into the formula. However, doing so would result in a division by zero, which is undefined in mathematics. Thus, according to the model, it's not possible to seize 100% of the drug due to an undefined cost.

Key Concepts

Cost Analysis

Cost Analysis is a critical tool in evaluating the necessary financial resources to achieve certain outcomes, like seizing drugs at a border, in this context. It helps us understand how costs change with different levels of action or intervention. In this problem, we use a formula, \( C = \frac{528p}{100-p} \), which relates the percentage \( p \) of drugs seized to the cost \( C \) in millions of dollars. This type of analysis allows decision-makers to allocate resources efficiently.

• For 25% seizure, the cost is \(176 million.
• For 50% seizure, the cost jumps to \)528 million.
• For 75% seizure, it becomes $1584 million.

These values highlight how costs escalate as a higher percentage of drugs are seized, demonstrating an increasing cost-benefit relationship. Additionally, attempting to seize 100% of the drugs leads to an undefined cost, which suggests that certain objectives may not be feasible with finite resources. This underscores the importance of cost analysis in planning and budgeting for real-world scenarios.

Percentage Calculations

Percentage Calculations help particularly in determining proportions of a quantity in terms of its total, expressed as a percentage. Here, the percentage \( p \) indicates how much of the drug is being seized relative to the total amount entering a country. Using the formula \( C = \frac{528p}{100-p} \), focus is on finding how the cost varies with each percentage point seized.

• To find the cost of a 25% seizure, use 25 for \( p \).
• For 50%, use 50; and for 75%, use 75 in the formula.

Each calculation involves basic algebraic substitution and simplification. This aligns with real-world scenarios where decisions are driven by budget constraints and objectives for percentage-based efficiency. Understanding how percentages affect variables like cost is a powerful tool in both math and economics.

Linear Equations

Linear Equations in this context, help explain relationships between two variables, often showing direct proportionality as seen in the cost formula provided: \( C = \frac{528p}{100-p} \). Although this formula is not linear throughout due to the division, it bears characteristics similar to linear equations in terms of direct relationships.
For linear understanding, consider that up to a certain threshold, cost increases as a fraction of \( p \). The more significant aspect, however, is the infinite nature of cost as \( p \) nears 100, showing a tendency of linear equations to illustrate limitation points and asymptotic behavior.
Comprehending such relationships enables better management of resources as it immediately tells us extremes such as the impracticality of a 100% target. This aligns with the principle of diminishing returns, where increases in \( p \) would result in ever greater costs, thus steering strategic decisions.