Call Now to Set Up Tutoring:

(888) 888-0446

Create an account to track your scores
and create your own practice tests:

Create an Account Create a Practice Test

All Math Modeling Resources

20 Practice Tests Question of the Day Flashcards Learn by Concept

Want to review Math Modeling but don’t feel like sitting for a whole test at the moment? Varsity Tutors has you covered with thousands of different Math Modeling flashcards! Our Math Modeling flashcards allow you to practice with as few or as many questions as you like. Get some studying in now with our numerous Math Modeling flashcards.

Flashcards: Math Modeling

A beauty supply company manufactures a variety of brushes. Quality control engineers work to ensure that the defected brushes in the factory will be detected prior to shipping them. It is estimated that approximately 0.2% of the brushes made will be defected. Tests can be done individually on the brushes or on batches of the brushes. If the test on a batch of brushes fails, it means that one or more of the brushes in that particular batch are defected. The estimated cost is 4 cents for a single brush, and 3+n cents for a group of n>1 brushes. If a batch fails then all brushes in that batch must be tested individually. Find the most cost-effected quality control procedure for detecting defected brushes.

By testing brush batches in groups of 15 will reduce testing costs without sacrificing the quality.

By testing brush batches in groups of 10 will reduce testing costs without sacrificing the quality.

By testing brush batches in groups of 9 will reduce testing costs without sacrificing the quality.

By testing brush batches in groups of 20 will reduce testing costs without sacrificing the quality.

By testing brush batches in groups of 12 will reduce testing costs without sacrificing the quality.

Report an Error Copyright Notice

Flashcards: Math Modeling

Your answer:

By testing brush batches in groups of 10 will reduce testing costs without sacrificing the quality.

Your answer:

By testing brush batches in groups of 9 will reduce testing costs without sacrificing the quality.

Your answer:

By testing brush batches in groups of 20 will reduce testing costs without sacrificing the quality.

Your answer:

By testing brush batches in groups of 12 will reduce testing costs without sacrificing the quality.

Correct answer:

By testing brush batches in groups of 15 will reduce testing costs without sacrificing the quality.

Explanation:

First identify the known variables and assumptions.

$\\n=\text{number of brushes per test batch} \\C=\text{testing cost for one group(cent)} \\A=\text{average testing cost (cents/brush)}$

$n=1\rightarrow A=4$

If a batch of brushes is tested and if the test shows that all the brushes are good then,

$n>1\rightarrow C=3+n$

If the batch test shows that there is a defected brush in the batch then,

$A=\text{(Average value of C)}/n$

Use a discrete probability model to find the most cost-effected quality control procedure for detecting defected brushes.

Consider the random variable

$X\epsilon\begin{Bmatrix} x_1, x_2, x_3,... \end{Bmatrix}$

that has a probability of

$\text{Pr}\begin{Bmatrix} X=x_i \end{Bmatrix}=p_i$

$\\\Sigma p_i=1 \\EX=\Sigma x_ip_i$

If the probability of a brush being good is then the probability of a brush being defected is 1-p . Then the average expected value of is as follows:

EC=(3+n)p+[(3+n)+4n](1-p)

Now there are brushes and the probability that one brush is defected is 0.002 thus assuming independence, the probability of all brushes in one test group are good is p=0.998^n .

Therefore the expected value of the random variable is,

$\\EC=(3+n)0.998^n+[(3+n)+4n](1-0.998^n) \\EC=(3+n)+4n(1-0.998^n) \\EC=3+5n-4n(0.998^n)$

Therefore the average testing cost is,

$A=\frac{3}{n}+5-4(0.998)^n$

Using the law of large numbers minimizing results in

A=1.31, n=15

Now answer the question.

By testing brush batches in groups of 15 will reduce testing costs without sacrificing the quality.

Flashcards: Math Modeling

There are two similar species of monkey that inhabit the same forest. The intrinsic growth rate is 4% for species A and 7% for species B. The capacity of the forest is estimated at 125,000 specie A monkeys and 450,000 species B monkeys. Poaching these monkeys over the last 100 years has caused for a decline in each species. Species A has a population of 4,000 and species B, a population of 80,000. Use a discrete model to project the population growth with a time step of several years. How large a time step can be used to maintain a continuous time model with an equivalent qualitative behavior?

Assume $\alpha =10^{-8}$ , x_b=100

h=47

h=57

h=37

h=30

h=50

Report an Error Copyright Notice

Flashcards: Math Modeling

Assume $\alpha =10^{-8}$ , x_b=100

Your answer:

h=57

Your answer:

h=37

Your answer:

h=30

Your answer:

h=50

Correct answer:

h=47

Explanation:

Identify the known information and the dynamical system.

$\\A=\text{number of species A monkey} \\B=\text{number of species B monkey} \\g_A=\text{growth rate of species A per year} \\g_B=\text{growth rate of species B per year} \\c_A=\text{competition on species A} \\c_B=\text{competition on species B} \\\\g_A=0.04A(1-A/125,000) \\g_B=0.07B(1-B/450,000) \\c_A=c_B=\alpha AB \\A\geq 0, b\geq 0 \\\alpha \epsilon \mathbb{R}>0$

$\\\frac{dx_1}{dt}=f_1(x_1,...,x_n) \\\vdots \\\\\frac{dx_n}{dt}=f_n(x_1,...,x_n)$

The dynamical system equation is:

$\frac{dx}{dt}=F(x)$

From here solve by using the Euler Method for a discrete time dynamical system.

$\frac{\Delta x}{\Delta t}=F(x)$

Now formulate the model

$\\\frac{dx_1}{dt}=f_1(x_1,x_2)=0.04x_1\left(1-\frac{x_1}{125,000} \right )-\alpha x_1x_2 \\\\\frac{dx_2}{dt}=f_1(x_1,x_2)=0.07x_1\left(1-\frac{x_1}{450,000} \right )-\alpha x_1x_2$

$\\x_1=\text{species A monkey} \\x_2=\text{species B monkey}$

Now transform these equations into a set of difference equations.

$\\\Delta x_1=f_1(x_1,x_2)\Delta t \\\Delta x_2=f_2(x_1,x_2)\Delta t$

$\\\Delta x_1= \text{change in the species A monkey population} \\\Delta x_2= \text{change in the species B monkey population}$

Assume

$\\\alpha =10^{-8} \\x_1(0)=4,000 \\x_2(0)=80,000\\N=50$

where represents the number of iterations the computer will use when implementing the Euler Method.

The time step $h=\Delta t$ will need to be changed and the model reformulated to calculate how large the time step can be before it doesn't resemble a continuous time model.

Using computer implementation of the Euler Method on the population of species B monkeys the following graph is produced.

Screen shot 2016 05 05 at 6.06.50 am

It is seen that h=47 is the largest time step before the function begins to have chaos and not represent the behavior of a continuous system.

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

View Tutors

Grace
Certified Tutor

University of Massachusetts-Dartmouth, Bachelor of Education, Portuguese. University of Massachusetts-Dartmouth, Masters in B...

View Tutors

James
Certified Tutor

St. Mary of the Plains College, Bachelors, Mathematics. Baker University, Masters, Business Administration and Management.

View Tutors

Constance
Certified Tutor

Loyola University-Chicago, Bachelors, Psychology. Sam Houston State University, Masters, Counseling Psychology.

Popular Subjects

Reading Tutors in Phoenix, Chemistry Tutors in Atlanta, Algebra Tutors in Phoenix, Computer Science Tutors in Boston, English Tutors in Denver, Algebra Tutors in San Diego, Reading Tutors in Miami, ACT Tutors in Boston, Biology Tutors in Boston, GRE Tutors in Philadelphia

Popular Courses & Classes

GMAT Courses & Classes in Boston, SAT Courses & Classes in Chicago, LSAT Courses & Classes in San Diego, GMAT Courses & Classes in San Diego, SSAT Courses & Classes in New York City, GRE Courses & Classes in Boston, Spanish Courses & Classes in Chicago, ACT Courses & Classes in Denver, MCAT Courses & Classes in Philadelphia, SSAT Courses & Classes in Seattle

Popular Test Prep

GMAT Test Prep in Philadelphia, GMAT Test Prep in Boston, ISEE Test Prep in Washington DC, SAT Test Prep in San Diego, ACT Test Prep in Denver, GRE Test Prep in Boston, SSAT Test Prep in San Diego, SAT Test Prep in Houston, SSAT Test Prep in Chicago, ISEE Test Prep in Los Angeles

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Create an account to track your scores
and create your own practice tests:

All Math Modeling Resources

Flashcards: Math Modeling

Flashcards: Math Modeling

Flashcards: Math Modeling

Flashcards: Math Modeling

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback:

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Create an account to track your scores and create your own practice tests:

All Math Modeling Resources

Flashcards: Math Modeling

Flashcards: Math Modeling

Flashcards: Math Modeling

Flashcards: Math Modeling

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Create an account to track your scores
and create your own practice tests: