application+of+first+order+_a01

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UNIVERSITI TUN HUSSEIN ONN MALAYSIA

FACULTY OF SCIENCE, TECHNOLOGY AND HUMAN DEVELOPMENT

SEMESTER 2 SESSION 2013/2014

BFC 14003/ BSM 10203

Assig!"# $ M%&'s (10)*Group work

APPLICATION OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

1. POPULATION

In a certain bacterial culture, the rate of population increase is proportional to the number of

bacteria present. Given that the initial population is 21!×

"i# If the number $ouble in three hours, fin$ the hourl% &ro'th.

"ii# (o' man% bacteria are there after ) hours*

"iii# (o' man% hours for the population becomes +1+× ?

2. POPULATION

A population of insects in a re&ion 'ill &ro' at a rate that is proportional to the current

population. In the absence of an% outsi$e factors, the population 'ill triple in t'o 'ees time.On an% &iven $a%, there is a net mi&ration into the area of 1- insects an$ 1! are eaten b% the bir$s

an$ $ie of natural causes. If there are initiall% 1 insects in the area,

"i# 'ill the population survive*"ii# If not, 'hen $o the% $ie out*

/. 0A3 4 (AL5 LI5

In ph%sics, the half life is a measure of the stabilit% of a ra$ioactive substance. The half life is

simpl% the time taen for one6half of the atoms in an initial amount A to $isinte&rate. The

lon&er the hal6life of a substance, the more stable it is.

A bree$er reactor converts the relativel% stable uranium 2/7 into the isotope plutonium 2/).

After 1- %ears, it is $etermine$ that .+/8 of the initial amount A of the plutonium has

1



A simple electric circuit consists of current I "ampere#, a resistance R "ohm# an$ an in$uctor L

"henr%#, also a constant electromotive force E "volts#. The current I satisfies the $ifferential

e>uation

E RI

dt

dI L =+

T'o )6volt batteries are connecte$ to a series in 'hich the in$uctance is D henr% an$ the

resistance is 7 ohms. 0etermine the current #"t I if the initial current is ?ero.

7. I

The &reat $etective Eohn 5lani&an an$ his assistant 0aphne Potts, are $iscussin& the mur$er of

the famous actress usan teele. Ciss teele 'as shot in the hea$ an$ her bo%frien$, Car9eese 'as foun$ stan$in& over the bo$% 'ith the mur$er 'eapon in han$ at 1.! p.m., at 'hich

time the coroner recor$e$ a bo$% temperature of .). F The room thermostat 'as set at

.2

F At 11.! p.m. the coroner too another rea$in& an$ recor$e$ a bo$% temperature of .!.-

F

Ciss teele;s normal temperature 'as F

!.)7 . 9eese 'as on sta&e bet'een !. p.m. until

7. p.m. 0etermine the time of mur$er an$ coul$ 9eese be claime$ as the prime suspect*

). TAFING A :9AF

I have Fit Fat an$ a mu& of hot Nescafe for m% lunch brea. The Nescafe is at .1) F

The room temperature is . F At time t , the Nescafe is coolin& at F

1- per minute.

"i# Co$el the e>uation of the coolin& Nescafe

"ii# (o' lon& $oes it tae for the temperature to reach .1+/ F

1. CON3 CON3 CON3

A person places 9C 2 in a savin& account 'hich pa%s - percent interest per %ear,

compoun$e$ continuousl%. (o' much mone% he has after three %ears* Also, $etermine the time

for the account to $ouble in value, presumin& no 'ith$ra'als an$ no a$$itional $eposits.

11. 5ALLING O:ET

3



An obBect 'ei&hin& !+ lbs is $roppe$ from a hei&ht of 1 ft 'ith an initial velocit% of 1ft@sec.

Assume the air resistance is proportional to the velocit% of the bo$%. If the limitin& velocit% is

no'n to be 127 ft@sec,

"i# 5in$ an e=pression for the velocit% of the obBect at an% time, t

"ii# 5in$ an e=pression for the position of the obBect at an% time,t

12. A CIHING P9O:LC

This mo$el concerns the flo' of a salt solution into a tan from 'hich the 'ell stirre$ mi=ture is

allo'e$ to flo' out. uppose initiall% that the tan contains 2 liters of solution containin& 2& of salt in solution. A solution containin& / &@ liter flo's in at a rate of 2 liters@min, an$ the

'ell6stirre$ mi=ture flo's out at the same rate. (o' man% ilo&rams of salt are in the tan at

time t *

Let#"t A

be the number of ilo&rams of salt in the tan at timet .

1/. 09OPPING :O03

A bo$% 'ei&hin& !+ & is $roppe$ from a hei&ht of 1 m 'ith an initial velocit% of 1m@sec.

Assume that the air resistance is proportional to the velocit% of the bo$%. If the limitin& velocit%

is no'n to be 127m@sec, fin$

"a# an e=pression for the velocit% of the bo$% at an% time, t

"b# an e=pression for the position of the bo$% at an% time, t

1+. 0A3

A certain radioactive material is known to decay at the rate proportional to the

amount present. If initially there is 100 milligrams of the material present and after

two hours it is oserved that the material has lost 10 percent of its original mass. !y

using growth population formula,dx

kxdt

= , "nd

#i$ an e%pression for the mass of the material remaining at any time t ,

#ii$ the mass of the material after "ve hours,

#iii$ the time at which the material has decayed to one half of its initial

mass.

&rite your answers in #ii$ and #iii$ in two decimal places.

1-. 9A0IOATI

'



A certain ra$ioactive material is no'n to $eca% at a rate proportional to the amount present. If

initiall% there is - milli&rams of the material present an$ after t'o hours it is observe$ that thematerial has lost 1 percent of its ori&inal mass, fin$

"a# an e=pression for the mass of the material remainin& at an% timet

"b# the mass of the material after four hours

"c# the time at 'hich the material has $eca%e$ to one half of its initial mass

1!. POPULATION

The rate of &ro'th of a certain animal population is $irectl% proportional to the population at

time, t "measure$ in %ears#, 'rite $o'n a $ifferential e>uation relatin& x an$ t . 5in$ the animal

population in %ear 2 base$ on the information &iven belo'.

NUC:9 3A9 POPULATION

1 1) /!7/ Thousan$s

2 1)7 ++-/ Thousan$s

1. POPULATION

In a stu$% on the effectiveness of a t%pe of an insect poison, it 'as foun$ that the rate of $ecrease

of the insect population is &iven b% , 'here is the time taen in hours after the

poison is a$ministrate$. Initiall%, there are - insects. 5in$J

"a# The number of insect left 2+ hours after a$ministration of poison.

"b# The time taen to $estro% half the insect population.

17. OOLING

Accor$in& to the Ne'ton;s La' of coolin&, the rate at 'hich a bo$% cools is proportional to the

$ifference bet'een the temperature of the bo$% an$ that of the surroun$in& me$ium. Let

represent the temperature "in K# of a bo$% in a room 'hose temperature is ept at a constant

(



17K. If the bo$% cools from K to -K in - minutes, ho' much lon&er 'ill it taes for its

temperature to $rop to +K*

1). OOLING

Un$er a certain con$ition, the rate of temperature $rop of a coolin& bo$% in a me$ium of a

constant temperature is proportional to the $ifference in temperature bet'een the bo$% an$ the

surroun$in& me$ium. Interpret this statement in the form of a $ifferential e>uation.

If the temperature $rop is from o to !o in / minutes 'hen the surroun$in& temperature is

1-o, fin$

"a# The temperature of the bo$% / minutes later,#$ The time taen for the bo$% to cool from !o to +o.

)

application+of+first+order+_a01

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