/*! 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 13 Draw a contour diagram for \(C(d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 13

Draw a contour diagram for \(C(d, m)=40 d+0.15 m\) Include contours for \(C=50,100,150,200\)

Short Answer

Expert verified
Plot lines for each contour equation for \( C = 50, 100, 150, 200 \) on a graph with \( d \) and \( m \) axes.

Step by step solution

01

Understand the Function

The given function is \( C(d, m) = 40d + 0.15m \). Here, \(d\) is likely representing the number of days and \(m\) the minutes. The function is linear, meaning it has a constant rate of change.
02

Set Up Contour Levels

To draw the contour diagram, we need to solve for \( m \) given specific values of \( C \). We have contour levels \( C = 50, 100, 150, 200 \). For each contour, we will solve \( C = 40d + 0.15m \) for \( m \).
03

Solve for Contour \( C = 50 \)

Set \( 40d + 0.15m = 50 \). Solve for \( m \):\[ 0.15m = 50 - 40d \]\[ m = \frac{50 - 40d}{0.15} \]
04

Solve for Contour \( C = 100 \)

Set \( 40d + 0.15m = 100 \). Solve for \( m \):\[ 0.15m = 100 - 40d \]\[ m = \frac{100 - 40d}{0.15} \]
05

Solve for Contour \( C = 150 \)

Set \( 40d + 0.15m = 150 \). Solve for \( m \):\[ 0.15m = 150 - 40d \]\[ m = \frac{150 - 40d}{0.15} \]
06

Solve for Contour \( C = 200 \)

Set \( 40d + 0.15m = 200 \). Solve for \( m \):\[ 0.15m = 200 - 40d \]\[ m = \frac{200 - 40d}{0.15} \]
07

Draw the Contour Diagram

Plot each contour by using the equations derived for \( m \) against \( d \). Each equation represents a line, and you should plot these lines on the same graph to create the contour diagram. The x-axis typically represents \( d \) (days), and the y-axis \( m \) (minutes).

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.

Linear Functions
Linear functions are a fundamental concept in mathematics characterized by a constant rate of change. These functions are often expressed in the form \( f(x) = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. The key properties include:
  • A linear graph is a straight line.
  • The slope \( m \) indicates the steepness and direction of the line.
  • The y-intercept \( b \) is where the line crosses the y-axis.
In the exercise, the linear function \( C(d, m) = 40d + 0.15m \) is given, which involves two variables: \( d \) for days and \( m \) for minutes. Here, the coefficients (40 and 0.15) represent the rate of change with respect to each variable.
Solving Equations
Solving equations often involves isolating the variable of interest. In our linear function, particular contour levels are given as \( C = 50, 100, 150, 200 \). The task is to solve for \( m \) when \( d \) is constant. Here are the steps to solve for \( m \):
  • Rearrange the equation to isolate \( m \) on one side, such as \( 0.15m = C - 40d \).
  • Divide both sides by 0.15 to solve for \( m \): \( m = \frac{C - 40d}{0.15} \).
By going through these steps for each contour level, you compute specific lines in terms of \( m \) and \( d \) that will help in plotting the contour diagram.
Graphing Techniques
Graphing is a crucial skill for visualizing functions and interpreting their relationships. Here are some basic graphing techniques:
  • Identify the variables: on a graph, typically one variable is plotted on the x-axis and another on the y-axis. For this exercise, \( d \) is the x-axis, and \( m \) is the y-axis.
  • Use a consistent scale: choose a reasonable scale that allows all data points and lines to fit clearly on the graph.
  • Plot each equation as a separate line: solve for different \( C \) values and plot their corresponding lines. In this case, for \( C = 50, 100, 150, 200 \).
  • Check the intersections and trends: since each equation depicts a contour level, observing how these lines interact can give insights into the solution's behavior.
Plotting these lines on the same graph results in a contour diagram, which effectively illustrates how changes in \( d \) and \( m \) affect \( C \)
Contour Levels
Contour levels in mathematics are a way to represent a three-dimensional surface on a two-dimensional plane through lines or curves. Each contour line corresponds to a specific value (level). In this case, each level represents a specific cost \( C \): 50, 100, 150, and 200.
  • Each contour line corresponds to a constant \( C \) value.
  • The spacing between the lines can indicate the rate of change; closer lines mean a steeper gradient.
  • Contour diagrams are commonly used in fields like meteorology, geography, and economics to represent concepts such as elevation changes, temperature changes, and cost changes.
By calculating equations for different contour levels and plotting them, you create a visual map of the function where each line represents a specific cost level \( C \). This helps in understanding how varying inputs \( d \) and \( m \) affect the resultant values of the function \( C(d, m) = 40d + 0.15m \).

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

Use Lagrange multipliers to find the maximum or minimum values of \(f(x, y)\) subject to the constraint. $$f(x, y)=x^{2}+3 y^{2}+100, \quad 8 x+6 y=88$$

Suppose that \(x\) is the price of one brand of gasoline and \(y\) is the price of a competing brand. Then \(q_{1},\) the quantity of the first brand sold in a fixed time period, depends on both \(x\) and \(y,\) so \(q_{1}=f(x, y) .\) Similarly, if \(q_{2}\) is the quantity of the second brand sold during the same period, \(q_{2}=g(x, y) .\) What do you expect the signs of the following quantities to be? Explain. (a) \(\partial q_{1} / \partial x\) and \(\partial q_{2} / \partial y\) (b) \(\partial q_{1} / \partial y\) and \(\partial q_{2} / \partial x\)

Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined. $$\frac{\partial A}{\partial h} \text { if } A=\frac{1}{2}(a+b) h$$

Find all the critical points and determine whether each is a local maximum, local minimum, a saddle point, or none of these. $$f(x, y)=x^{3}-3 x+y^{3}-3 y$$

A manufacturer sells two products, one at a price of \(\$ 4000\) a unit and the other at a price of \(\$ 13,000\) a unit. A quantity \(q_{1}\) of the first product and \(q_{2}\) of the second product are sold at a total cost of \(\$\left(4000+q_{1}+q_{2}\right)\) to the manufacturer. (a) Express the manufacturer's profit, \(\pi\), as a function of \(q_{1}\) and \(q_{2}\) (b) Sketch contours of \(\pi\) for \(\pi=10,000, \pi=20,000\) and \(\pi=30,000\) and the break-even curve \(\pi=0\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.