7.5 Challenge Worksheet

Transcription

7.5 Challenge Worksheet
Name
LESSON
7-5
Date
Class
Challenge
Exponential Heating and Cooling
Newton’s Law of Cooling states that the rate of heat loss
of an object is proportional to the difference in temperatures
between the object and its surrounding ambient temperature.
T t TA
This phenomenon is modeled with a differential equation
t TA
T
and that equation may be solved to give
_________
T t ⫽ T A ⫹ [ T 0 ⫺ T A] b t
where T t is the varying temperature of the object at a
given time, t, T A is the surrounding ambient temperature,
T 0 is the initial temperature of the object, and b is a
constant that depends on the material the object is
composed of and how fast it heats or cools.
Suppose you decided to make a cup of hot chocolate
heated to 180°F in the kitchen that is at 72°F.
1. Solve the above equation for the constant b.
2. If the cup of hot chocolate cooled to 150°F in
15 minutes, find the value of the constant b in
the above equation. Express your answer to
five decimal places.
3. Solve the above equation for t.
4. Suppose you like your hot chocolate at the tepid
temperature of 120°F. How long, to the nearest
minute, will you have to wait until it cools to this
temperature?
T 0 T A b t
bt
T0 TA
T t TA
t log b
log _________
T0 TA
T t TA
log _________
T0 TA
______________
log b
t
10
log
TT t TT A
_________
0
A
______________
t
b
b ; 0.97854
First 3 steps same as #1;
t TA
T
__________
T0 TA
log ____________
t
log b
About 37 min
To go along with your hot chocolate, you take a frozen cherry pie from the
freezer and place it in the oven preheated to 350°F. Assume the freezer is
at 32°F.
5. If the cherry pie comes to a temperature of 120°F in
20 minutes, find the value of the constant b in the
above equation. Express your answer to 5 decimal
places.
b ; 0.98362
6. How long will it take for the pie to reach its final
temperature of 220°F?
About 55 min
7. The pie is taken out of the oven and set on a table
in a room at 80°F. In 10 minutes it has cooled to
185°F. However, the pie must cool to 125°F
before it is ready to eat. How much longer will
you have to wait?
About 30 min
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a207c07-5_ch.indd 8
40
Holt Algebra 2
12/29/05 8:01:55 PM
Process Black
,%33/.
LJx
,iÌi>V…
%XPONENTIALAND,OGARITHMIC%QUATIONSAND)NEQUALITIES
CONTINUED
…>i˜}i
%XPONENTIAL(EATINGAND#OOLING
--"
LJx
.EWTONS,AWOF#OOLINGSTATESTHATTHERATEOFHEATLOSS
OFANOBJECTISPROPORTIONALTOTHEDIFFERENCEINTEMPERATURES
BETWEENTHEOBJECTANDITSSURROUNDINGAMBIENTTEMPERATURE
T
T E
4HISPHENOMENONISMODELEDWITHADIFFERENTIALEQUATION 4 T 4 ! 4 4 ! B 4
TTE4!
T
ANDTHATEQUATIONMAYBESOLVEDTOGIVE
?????????
!LOGARITHMICEQUATIONCONTAINSALOGARITHMICEXPRESSIONTHATHASAVARIABLE
LOGXISALOGARITHMICEQUATION
XSINCE
4STD4!QÊ44!RB
T
#OMBINEANDUSEPROPERTIESOFLOGARITHMSTOSOLVELOGARITHMICEQUATIONS
WHERE4STDISTHEVARYINGTEMPERATUREOFTHEOBJECTATA
GIVENTIMET4!ISTHESURROUNDINGAMBIENTTEMPERATURE
4ISTHEINITIALTEMPERATUREOFTHEOBJECTANDBISA
CONSTANTTHATDEPENDSONTHEMATERIALTHEOBJECTIS
COMPOSEDOFANDHOWFASTITHEATSORCOOLS
3OLVELOGXLOG
3TEP 5SETHE1UOTIENT0ROPERTYOF,OGARITHMS
LOGXLOG
LOG????
X
3TEP 3IMPLIFY
LOG????
X
LOGX
LOGXLOGYLOG??
XY
3UPPOSEYOUDECIDEDTOMAKEACUPOFHOTCHOCOLATE
HEATEDTO &INTHEKITCHENTHATISAT &
3OLVETHEABOVEEQUATIONFORTHECONSTANTB
3TEP 5SETHEDEFINITIONOFTHELOGARITHM
X
IFB ATHENLOGBAX
X
3TEP 3OLVEFORX$IVIDEBOTHSIDESBY
LOGLOGSXD
LOGXLOG
LOGX
LOGSXD
LOGX??
LOGSXD
X
LOG ????
X
LOGX
X
X
X
X
X
X
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
T E
X
X??
(OLT!LGEBRA
7HILE*OHNAND#ODYPLAYTHEIRFAVORITEVIDEOGAME
*OHNDRINKSCUPSOFCOFFEEANDACOLAAND#ODY
DRINKSCUPSOFBREWEDTEAANDACUPOFICEDTEA
*OHNRECALLSREADINGTHATUPTOMGOFCAFFEINEIS
CONSIDEREDAMODERATELEVELOFCONSUMPTIONPERDAY
4HERATEATWHICHCAFFEINEISELIMINATEDFROMTHE
BLOODSTREAMISABOUTPERHOUR
*OHNWANTSTOKNOWHOWLONGITWILLTAKEFORTHE
CAFFEINEINHISBLOODSTREAMTODROPTOAMODERATE
LEVEL
#AFFEINE#ONTENTOF3OME
"EVERAGES
"EVERAGE
#AFFEINE
MGPER
SERVING
"REWEDCOFFEE
"REWEDTEA
)CEDTEA
#OLA
LœÕÌÊÎäʓˆ˜
YANDYTET
#HOOSETHELETTERFORTHEBESTANSWER
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
)NSOLVINGEQUATIONSWITHLOGARITHMSANDEXPONENTSFIRSTUSETHEPROPERTIES
OFLOGARITHMSANDEXPONENTIALFUNCTIONSTOSIMPLIFYEQUATIONS(EREARETWO
ADDITIONALPROPERTIESTHATAREUSEFULFORSOLVINGEQUATIONS
s )FXYTHENBXBY
s )FXYTHENLOGBXLOGBY
ÊÓÊ{Ê
B 2EWRITETHEEQUATIONSOBOTHSIDESHAVE
THESAMEBASE7HATISTHEVALUEOFX
C 3HOWHOWYOUCANCHECKYOURSOLUTION
ÊÓÊXÊÊÊÓÊ{ÊÆÊXÊÊ{
ÊiÌ̈˜}ÊXÊÊ{]ÊÊÓÊ{ÊÊ£È
A 2EWRITETHEEQUATIONUSINGTHEDEFINITIONOFLOGARITHM
B 7HATISTHESOLUTIONOFTHEEQUATION
C 7HATISTHEVALUEOFLOG
X
X
Ê£äÊÊÓÊÊX
XÊÊ£ää
ʏœ}Ê£äÊ£ääÊÊʏœ}Ê£äÊ£äÊÊÓÊÊÓ
5SETHEEQUATION s FOR%XERCISE
A 2EWRITETHEEQUATIONSOTHATTHEEXPONENTSON
BOTHSIDESHAVETHESAMEBASE
Y
X
Y
B 3IMPLIFYUNTILITISINTHEFORM C 3OLVEFORX
Ó
ÊÎÊxÊÊÊÎÊXÊsÊÊÊTÊÎÊÓÊEÊ ÊÊÊÎÊxÊÊÊÎÊXÊsÊÊÎÊ{Ê
ÊÎÊxÊÊÊÎÊXÊÊ{Ê
xÊÊXÊÊ{ÆÊXÊÊ£°
5SETHEEQUATIONXLOGTXE LOGXFOR%XERCISE
A $ESCRIBEEACHSTEPINTHETABLETOSOLVETHEEQUATION
XLOGXLOGX
X
( HOURS
* HOURS
,i>`ˆ˜}Ê-ÌÀ>Ìi}Þ
5SE2ELATIONSHIPS
" HOURS
# HOURS
$ HOURS
œÌʏ}iLÀ>ÊÓ
5SETHEEQUATIONLOGXFOR%XERCISE
)F*OHNDRANKCUPSOFCOFFEEANDA
COLAABOUTHOWLONGWOULDITTAKEFOR
THELEVELOFCAFFEINEINHISSYSTEMTODROP
TOAMODERATELEVEL
& HOUR
' HOURS
!BOUTHOWLONGWOULDITTAKEFORTHE
LEVELOFCAFFEINEIN#ODYSSYSTEMTO
DROPBYAFACTOROF
! HOUR
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
H
C 3KETCHTHERESULTINGGRAPH
AK4up.indd 76
LœÕÌÊxxʓˆ˜
X
(ECANGRAPHTHEEQUATIONTETANDFINDTHEVALUEOFT
WHERE#TTEIS
B 7HATEQUATIONSDID#ODYENTERINTO
HISCALCULATOR
E
4HEPIEISTAKENOUTOFTHEOVENANDSETONATABLE
INAROOMAT &)NMINUTESITHASCOOLEDTO
&(OWEVERTHEPIEMUSTCOOLTO &
BEFOREITISREADYTOEAT(OWMUCHLONGERWILL
YOUHAVETOWAIT
A %XPRESSASAPOWEROF
A #ODYTHINKSTHATITWILLTAKEATLEASTHOURSFORTHELEVELOFCAFFEINEIN
*OHNSSYSTEMTODROPTOTHESAMELEVELOFCAFFEINETHAT#ODYCONSUMED
%XPLAINHOWHECANUSEHISGRAPHINGCALCULATORTOPROVETHAT
T
5SETHEEQUATION FOR%XERCISE
#TTE#TET
C (OWLONGTOTHENEARESTTENTHOFANHOURWILLIT
TAKEFORTHECAFFEINEIN*OHNSSYSTEMTOREACH
AMODERATELEVEL
BÊÊ䰙Çnx{
ˆÀÃÌÊÎÊÃÌi«ÃÊÃ>“iÊ>Ãʛ£Æ
4ÊÊÊTTEÊÊÊ4Ê!
Ê ÊÚÚÚÚÚÚÚÚÚÚ
Ê
Ê4ÊÊÊÊ4Ê!
ÚÚÚÚÚÚÚÚÚÚÚÚ
œ}ÊÊÊ
Ê
ÊÊT
œ}ÊB
LœÕÌÊÎÇʓˆ˜
BÊÊ䰙nÎÈÓ
LJx
MG
B 7RITEANEQUATIONSHOWINGTHEAMOUNTOFCAFFEINE
INTHEBLOODSTREAMASAFUNCTIONOFTIME
E
B
(OWLONGWILLITTAKEFORTHEPIETOREACHITSFINAL
TEMPERATUREOF &
--"
A (OWMUCHCAFFEINEDID*OHNCONSUME
E
)FTHECHERRYPIECOMESTOATEMPERATUREOF &IN
MINUTESFINDTHEVALUEOFTHECONSTANTBINTHE
ABOVEEQUATION%XPRESSYOURANSWERTODECIMAL
PLACES
*ÀœLi“Ê-œÛˆ˜}
%XPONENTIALAND,OGARITHMIC%QUATIONSAND)NEQUALITIES
LJx
T
4OGOALONGWITHYOURHOTCHOCOLATEYOUTAKEAFROZENCHERRYPIEFROMTHE
FREEZERANDPLACEITINTHEOVENPREHEATEDTO &!SSUMETHEFREEZERIS
AT &
3OLVEANDCHECK
LOGX T
4 TTE 4!
LOG ?????????
4 4!
??????????????
T
3UPPOSEYOULIKEYOURHOTCHOCOLATEATTHETEPID
TEMPERATUREOF &(OWLONGTOTHENEAREST
MINUTEWILLYOUHAVETOWAITUNTILITCOOLSTOTHIS
TEMPERATURE
X
??
X
T
3OLVETHEABOVEEQUATIONFORT
,%33/.
T
)FTHECUPOFHOTCHOCOLATECOOLEDTO &IN
MINUTESFINDTHEVALUEOFTHECONSTANTBIN
THEABOVEEQUATION%XPRESSYOURANSWERTO
FIVEDECIMALPLACES
2EMEMBER5SEASTHEBASE
WHENTHEBASEISNOTGIVEN
LOGX
E
E
B 44!
4TTE4!
?????????
LOG TLOGB
44!
TTE 4!
4
LOG ?????????
44!
??????????????
LOGB
T
(OLT!LGEBRA
SXLOGXLOGXD
5SEOFTHE0OWER0ROPERTY
XLOGXLOGX
>V̜ÀʏivÌÊÈ`i
`ˆÛˆ`iÊLœÌ…ÊÈ`iÃÊLÞÊÓ
S D
ÕÃiʜvÊ̅iÊ+՜̈i˜ÌÊ*Àœ«iÀÌÞ
X XLOG ????
X B 3IMPLIFYANDSOLVETHERESULTINGEQUATION
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
76
ÓXÊÊ£ÊÊxÆÊXÊÊÓ
œÌʏ}iLÀ>ÊÓ
Holt Algebra 2
1/4/06 3:44:25 PM