/*! 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 23 Suppose that the dollar cost of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 23

Suppose that the dollar cost of producing \(x\) washing machines is \(c(x)=2000+100 x-0.1 x^{2}\). a. Find the average cost per machine of producing the first 100 washing machines. b. Find the marginal cost when 100 washing machines are produced. c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

Short Answer

Expert verified
a) The average cost per machine is $110. b) The marginal cost at 100 machines is $80. c) The 101st machine costs approximately $79.9 to produce.

Step by step solution

01

Understand Average Cost Formula

The average cost per machine when producing a certain number of machines can be found using the formula \( \text{Average Cost} = \frac{c(x)}{x} \). We will use this to find the average cost per machine for the first 100 washing machines.
02

Calculate Average Cost for 100 Machines

Using the average cost formula: \( c(x) = 2000 + 100x - 0.1x^2 \). For 100 machines, substitute \( x = 100 \): \[ c(100) = 2000 + 100(100) - 0.1(100)^2 = 2000 + 10000 - 1000 = 11000 \] The average cost is \( \frac{11000}{100} = 110 \).
03

Understand Marginal Cost Concept

The marginal cost is the derivative of the cost function \( c(x) \), which represents the rate of change of cost with respect to quantity. It tells us the approximate cost of producing one additional machine.
04

Differentiate the Cost Function

Differentiate \( c(x) = 2000 + 100x - 0.1x^2 \) with respect to \( x \). The derivative is: \[ c'(x) = 100 - 0.2x \].
05

Calculate Marginal Cost for 100 Machines

Substitute \( x = 100 \) into the marginal cost formula: \[ c'(100) = 100 - 0.2(100) = 100 - 20 = 80 \]. So, the marginal cost when 100 machines are produced is \( 80 \).
06

Calculate Cost to Produce One More (101st) Machine

To find the cost of producing the 101st machine, calculate the original cost with 101 machines and subtract the cost with 100 machines: \[ c(101) = 2000 + 100(101) - 0.1(101)^2 = 2000 + 10100 - 1020.1 = 11079.9 \]Subtract \( c(100) = 11000 \): \( 11079.9 - 11000 = 79.9 \).
07

Compare Marginal Cost and Additional Cost

The marginal cost at 100 machines is 80, and the cost of producing the 101st machine is approximately 79.9. These values are very close, verifying that the marginal cost is a good approximation of the cost to produce one additional machine.

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.

Average Cost Calculation
Calculating the average cost is essential when trying to understand how much each unit of a product contributes to total production costs. For example, in producing washing machines, knowing the average cost helps in pricing and profitability analysis. The average cost is determined using the formula:
  • \( \text{Average Cost} = \frac{c(x)}{x} \)
This represents the total cost \( c(x) \) divided by the number of units produced \( x \). By using this, you capture the overall cost spread over each product unit. Consider the production of 100 washing machines: the formula gives you the total cost of \( 11000 \) divided by 100, leading to an average cost of \( 110 \) per machine.
Understanding average costs helps businesses decide how many units to produce and the optimal scale of operation. Sometimes, knowing the average can inform strategic decision-making and identify whether or not economies of scale are being realized.
Marginal Cost Importance
Marginal cost plays a crucial role in manufacturing and economics. It represents the cost incurred by producing one additional unit of a product. Understanding this cost can significantly influence production decisions. It鈥檚 calculated by taking the derivative of the cost function, indicating how the total cost changes with each unit produced:
  • For washing machines, the marginal cost is found by the derivative \( c'(x) = 100 - 0.2x \).
When you substitute 100 into this, the marginal cost is 80, showing that producing the 101st machine will approximately cost an additional \( 80 \).
Knowing the marginal cost helps businesses decide whether producing additional units will be profitable. If the selling price per unit exceeds the marginal cost, then increasing production is beneficial.
It also aids in setting competitive pricing strategies, staying responsive to market dynamics, and optimizing resource allocation.
Derivative in Economics
The derivative is a powerful mathematical tool used extensively in economics to analyze functions and understand how changes in one quantity affect another. In cost analysis, derivatives show how cost changes with production levels, helping firms predict expenses and adjust strategies accordingly.
  • For example, the derivative of the cost function \( c(x) = 2000 + 100x - 0.1x^2 \) is \( c'(x) = 100 - 0.2x \).
This derivative reveals the marginal cost, indicating the rate at which total costs change as output increases.
By knowing the derivative, businesses can make informed decisions about scaling production, anticipate financial requirements, and minimize wastage.
Understanding derivatives helps in differentiating between fixed and variable cost components and optimizing the overall cost structure.

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

Two ships are steaming straight away from a point \(O\) along routes that make a \(120^{\circ}\) angle. Ship \(A\) moves at 14 knots (nautical miles per hour; a nautical mile is \(1852 \mathrm{m}\) ). Ship \(B\) moves at 21 knots. How fast are the ships moving apart when \(O A=5\) and \(O B=3\) nautical miles?

Assume that a particle's position on the \(x\) -axis is given by $$x=3 \cos t+4 \sin t$$ where \(x\) is measured in meters and \(t\) is measured in seconds. a. Find the particle's position when \(t=0, t=\pi / 2,\) and \(t=\pi\) b. Find the particle's velocity when \(t=0, t=\pi / 2,\) and \(t=\pi\)

A particle moves along the parabola \(y=x^{2}\) in the first quadrant in such a way that its \(x\) -coordinate (measured in meters) increases at a steady \(10 \mathrm{m} / \mathrm{s}\). How fast is the angle of inclination \(\theta\) of the line joining the particle to the origin changing when \(x=3 \mathrm{m} ?\)

Suppose that a piston is moving straight up and down and that its position at time \(t\) s is $$s=A \cos (2 \pi b t)$$ with \(A\) and \(b\) positive. The value of \(A\) is the amplitude of the motion, and \(b\) is the frequency (number of times the piston moves up and down each second). What effect does doubling the frequency have on the piston's velocity, acceleration, and jerk? (Once you find out, you will know why some machinery breaks when you run it too fast.)

Suppose that functions \(f\) and \(g\) and their derivatives with respect to \(x\) have the following values at \(x=2\) and \(x=3\) $$\begin{array}{ccccc}\hline x & f(x) & g(x) & f^{\prime}(x) & g^{\prime}(x) \\\\\hline 2 & 8 & 2 & 1 / 3 & -3 \\ 3 & 3 & -4 & 2 \pi & 5 \\\\\hline\end{array}$$ Find the derivatives with respect to \(x\) of the following combinations at the given value of \(x\) a. \(2 f(x), \quad x=2\) b. \(f(x)+g(x), \quad x=3\) c. \(f(x) \cdot g(x), \quad x=3\) d. \(f(x) / g(x), \quad x=2\) e. \(f(g(x)), \quad x=2\) f. \(\sqrt{f(x)}, \quad x=2\) g. \(1 / g^{2}(x), \quad x=3\) h. \(\sqrt{f^{2}(x)+g^{2}(x)}, \quad x=2\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.