/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 The cost \(C\) (in millions of d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 63

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

Expert verified
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

01

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 \(p=75\), \(C=528*75/(100-75) = \$1584M\)
02

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, \(\lim_{p \to 100^-}C = \infty\)
03

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.
04

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find an equation of the tangent line to the function at the given point. Then find the function values and the tangent line values at \(f(x+\Delta x)\) and \(y(x+\Delta x)\) for \(\Delta x=-0.01\) and \(0.01\). \(f(x)=\sqrt{25-x^{2}}\) \((3,4)\)

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=\frac{x^{2}+1}{x^{2}-9}\)

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=\frac{2 x}{x^{2}-1}\)

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=-4 x^{3}+6 x^{2}\)

The cost \(C\) (in dollars) of producing \(x\) units of a product is \(C=1.35 x+4570\) (a) Find the average cost function \(\bar{C}\). (b) Find \(\bar{C}\) when \(x=100\) and when \(x=1000\). (c) What is the limit of \(\bar{C}\) as \(x\) approaches infinity?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.