/*! 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 15 The cost of controlling emission... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 15

The cost of controlling emissions at a firm is given by $$C(q)=4,000+100 q^{2}$$ where \(q\) is the reduction in emissions (in pounds of pollutant per day) and \(C\) is the daily cost to the firm (in dollars) of this reduction. Government clean-air subsidies amount to $$\$ 500$$ per pound of pollutant removed. How many pounds of pollutant should the firm remove each day in order to minimize \(n e t\) cost (cost minus subsidy)?

Short Answer

Expert verified
To minimize the net cost, the firm should remove \(\frac{5}{2}\) pounds of pollutant per day.

Step by step solution

01

Find the net cost function

We're given the cost function as \(C(q)=4,000+100q^2\). We know the subsidy per pound of pollutant removed is \(500\). So, the net cost function can be written as: \[N(q) = C(q) - 500q\] Substitute the given cost function into this equation: \[N(q) = (4,000 + 100q^2) - 500q\]
02

Take the derivative of the net cost function

To find the minimum of the net cost function, we will take the derivative with respect to \(q\) and set it equal to zero. So, we have: \[\frac{dN(q)}{dq} = \frac{d(4,000 + 100q^2 - 500q)}{dq}\] Applying the differentiation rules, we get: \[\frac{dN(q)}{dq} = 200q - 500\]
03

Set the derivative equal to zero and solve for \(q\)

To find the minimum value, set the derivative equal to zero and solve for \(q\): \[200q - 500 = 0\] Now, solve for \(q\): \[q = \frac{500}{200} = \frac{5}{2}\] So the firm should remove \(q = \frac{5}{2}\) pounds of pollutant per day to minimize the net cost.
04

Conclusion

The firm should remove \(\frac{5}{2}\) pounds of pollutant per day in order to minimize the net cost (cost minus subsidy).

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.

Understanding Cost Functions
A cost function is essentially a formula that represents the cost associated with producing a specific quantity of goods or, in this case, reducing a certain level of pollution. It helps firms forecast expenses based on various factors. In our example, the cost function is \(C(q) = 4,000 + 100q^2\). Here, \(q\) represents the amount of emissions reduced in pounds per day.
  • The constant \(4,000\) is a fixed cost, which means it's incurred regardless of the level of emissions reduced.
  • The term \(100q^2\) signifies the variable cost that changes with the amount of emissions controlled. The \(q^2\) suggests this is a nonlinear cost function where costs increase quadratically as more pollutants are reduced.

This function illustrates how financial decisions are influenced by both fixed and variable costs, giving insight into the overall expenses tied to emission control activities.
Impact of Subsidies on Costs
Subsidies are financial aids provided by the government to reduce the burden on firms. In this scenario, the government offers a subsidy of \$500 per pound of pollutant removed.
This type of subsidy plays a crucial role in reducing the net cost calculated by firms when they decide the amount of emissions to remove.
  • A subsidy effectively lowers the firm's total expenses by a fixed rate per unit of output or action, like reducing pollution.
  • It encourages firms to engage more in activities recommended by governmental policies, such as reducing emissions, to benefit from subsidies.

In the exercise, the subsidy helps calculate the net cost, which is given by the formula \(N(q) = C(q) - 500q\). This enabled the firm to calculate how much it truly costs after accounting for the government incentives while optimizing their pollutant reduction strategies.
Role of Derivatives in Optimization
Derivatives are mathematical tools used to measure how a function changes as its input changes. In optimization problems, they help in finding maximum or minimum values of functions. Here, we apply derivatives to the net cost function \(N(q) = C(q) - 500q\) to determine the optimal reduction level.
  • The derivative \(\frac{dN(q)}{dq}\) of the net cost function indicates the rate of change of costs with respect to emissions reduction.
  • To find the point where costs are minimized, set the derivative equal to zero. This gives the point where the slope of the curve is flat, indicating a local minimum or maximum.

From this, we derived \(200q - 500 = 0\). Solving for \(q\), we found that removing \(\frac{5}{2}\) pounds of pollutant per day will minimize net costs. This process illustrates how derivatives are vital in optimizing business and economic decisions by finding points of cost minimization.

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

The following graph shows the approximate value of the U.S. Consumer Price Index (CPI) from September 2004 through November \(2005.32\) The approximating curve shown on the figure is given by $$I(t)=-0.005 t^{3}+0.12 t^{2}-0.01 t+190 \quad(0 \leq t \leq 14)$$ where \(t\) is time in months ( \(t=0\) represents September 2004). a. Use the model to estimate the monthly inflation rate in July \(2005(t=10)\). [Recall that the inflation rate is \(\left.I^{\prime}(t) / I(t) .\right]\) b. Was inflation slowing or speeding up in July \(2005 ?\) c. When was inflation speeding up? When was inflation slowing? HINT [See Example 3.]

If the graph of a function has a horizontal asymptote at \(y=a\) in such a way that \(y\) decreases to \(a\) as \(x \rightarrow+\infty\), what can you say about the graph of its derivative? Explain.

Sketch the graph of the given function, indicating (a) \(x\) - and \(y\) -intercepts, (b) extrema, (c) points of inflection, \((d)\) behavior near points where the function is not defined, and (e) behavior at infinity. Where indicated, technology should be used to approximate the intercepts, coordinates of extrema, and/or points of inflection to one decimal place. Check your sketch using technology. \(f(x)=x^{2}+2 x+1\)

Your other friend tells you that she has found a continuous function with two critical points, one a relative minimum and one a relative maximum, and no point of inflection between them. Can she be right?

Rewrite the statements and questions in mathematical notation. There are presently 400 cases of Bangkok flu, and the number is growing by 30 new cases every month.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.