Description

Hello,

I need help solving this assignment, I ran out of time and I need someone to help me solve this math assignment.Thank you

1 attachmentsSlide 1 of 1attachment_1attachment_1

Unformatted Attachment Preview

Department of Mathematics UAB

Differential Equations

MA252 Fall 2020

ASSIGNMENT 3

Due date: Tuesday 09/15/20

Applications of First Order Differential Equations

Your report on this assignment should conform to the requirements given in the information

sheet; in particular the questions (and parts thereof) are to be answered in increasing order

with MAPLE calculations alternating with relevant commentary (no “MAPLE appendices”

permitted). You should describe fully how you solved these modeling problems. Use the text

insertion capabilities of MAPLE where appropriate; in particular you should add text briefly

describing each MAPLE command before you execute it. The MAPLE file ass3-fall20.mw

should be downloaded from Canvas and used as a command template.

1. A Chemical Reaction.

Two chemicals X and Y react in such a way that one gram of X combines with 4

grams of Y to produce a compound Z which is written by the chemists as

X + 4Y → Z.

The law of mass action in chemical kinetics states that, at a given time t, the rate

at which X and Y react (i.e. the rate at which Z is produced) is proportional to the

product of the amounts of X and Y that remain untransformed at that time (so this is

a second order reaction). Assume that there are initially 50 grams of X and 32 grams

of Y , and it is known that 30 grams of Z are formed in 10 minutes.

(a) Let z(t) grams of Z be present at time t. Derive a differential equation initial

value problem for z(t) and solve it by hand calculation.

(b) Check your solution in (a) using MAPLE.

(c) How much of the compound Z is present after 15 minutes?

(d) How much Z is present after a very long time (t → ∞)?

2. The Corporate Boardroom.

A corporate boardroom contains 4500 ft3 of air initially free of carbon monoxide. Each

morning (time t = 0) the executives arrive and cigarette smoke containing 4% carbon

monoxide by volume is introduced into the room at the rate of 0.3 ft3 /min by the

smokers. A ceiling fan keeps the air in the room well circulated, and the air and smoke

mixture leaves the room at the rate of 10 ft3 /min. After exiting the room the carbon

monoxide is removed from the air and smoke mixture via a catalytic converter system

before being returned as clean air to the room at the same rate by the building’s

forced-air system.

Figure 1: The corporate board room

(a) Explain in detail how to find a linear differential equation for the number of cubic

feet, A(t), of carbon monoxide in the room at time t minutes. Using the initial

condition A(0) = 0, solve this equation by hand to determine a formula for A(t).

(b) Check your result in (a) by solving the initial value problem using MAPLE.

(c) Use the MAPLE plot() command to plot the carbon monoxide concentration over

the t range [0, 10000]. Observe that, to activate the solution A(t) as a MAPLE

function (and avoid ‘‘empty plot’’ errors), you will need to use the MAPLE

unapply() and rhs() functions as we did in class. From your solution A(t) use

the MAPLE fsolve() command to find how long before the concentration of

carbon monoxide becomes 0.02%.

(d) It is known that a carbon monoxide concentration of 0.16% is lethal. Can such

life-threatening concentrations appear in the conference room? It is also known

that exposure to concentrations at or above 0.02% are associated with loss of

judgement on the part of the subjects. Since the occupants of the room are the

senior executives in the company, do you have any recommendations for savvy

investors holding stock in the company?

3. Linear First Order Differential Equation

The general solution of a first-order differential equation is a formula, containing one

arbitrary constant, representing all solutions of the equation. Find, by hand, showing

all your working, the general solution of the first-order linear equation

(x2 − 1)

dy

− xy = 2x(x2 − 1).

dx

Hint: be careful, this equation is not yet in “standard form”.

Purchase answer to see full

attachment

Tags:

chemical reaction

Carbon dioxide

general solution

MAPLE calculations

orporate Boardroom

User generated content is uploaded by users for the purposes of learning and should be used following Studypool’s honor code & terms of service.

## Reviews, comments, and love from our customers and community:

This page is having a slideshow that uses Javascript. Your browser either doesn't support Javascript or you have it turned off. To see this page as it is meant to appear please use a Javascript enabled browser.