(Effective Alternative Secondary Education)
MATHEMATICS II Y
X
MODULE 3
Variation
BUREAU OF SECONDARY S ECONDARY EDUCAT EDUCATION Department of Education DepEd Complex, eralco A!enue, A!enue, "a#i$ Cit%
&
Module 3 Variation What this module is about This module module deals deals with with the variation variation of more more than two variables. variables. oth direct and inverse variations variations may occur occur in the same !roblem. "oint "oint variation are #uantities #uantities that are directly related. related. ut when $oint variation is combined with inverse variation% then it is called combined variations.
What you are expected to learn &. identify relationshi! relationshi! involvin' two or more variables. variables. . find the the relati relation on and and constan constantt of variat variation ion . a!!ly a!!ly the conce conce!t !t of !ro!o !ro!ortio rtional nality ity..
How much do you know A. *sin' + as the constant constant of of variation% variation% write write the e#uation e#uation of variation variation for for each of of the followin', &. The area (A) of a !arallelo'ram varies $ointly as its base (b) and its altitude (a). . The volume (-) of a !yramid varies $ointly as its base area (b) and its altitude (a). . The area of the circle varies directly as the s#uare of its radius. . * varies $ointly as the s#uare of m and inversely as n. /. - varies $ointly with l % w and h. 0. The volume ( V ) of a cube varies directly as the cube of its ed'e ( e). 1. The force ( F ) needed to !ush an ob$ect alon' a flat surface varies directly as the wei'ht ( w ) of the ob$ect. 2. The altitude ( h) of a cylinder is inversely !ro!ortional to the s#uare of its radius ( r ). ). '
3. M varies directly as r and inversely as s. &4. Q varies $ointly as R and T .
What you will do 5esson & "oint -ariation This lesson deals with another conce!t of variation% the $oint variation. Some !hysical relationshi!s% as in area or volume% may involve three or more variables simultaneously. Consider the area of a rectan'le which is obtained from the formula A lw where l is the len'th w is the width of the rectan'le. The table shows the area in s#uare centimetres for different values of the len'th and the base. =
l w
A
0
&
/ &/
0 / 4
0 1
2 1 /0
2 && 22
&4 & &4
6bserve that A increases as either l or w increase or both. Then it is said that the area of a rectan'le varies $ointly as the len'th and the width. Consider the area of a trian'le% which is obtained from the formula,
A
& =
ab
'
where b is the base and a is the altitude of the trian'le. The table shows the area in s#uare centimetres for different values of the base and altitude% both bein' in centimetres. b a A
0
/ &4
0 / &/
0 1 &
2 1 2
2 &&
&4 & 0/
6bserve that A increases as either b or a increase or both. 7e say that the area of a trian'le varies $ointly as the base and the altitude.
Examples:
(
&. 8ind an e#uation of variation where b ( and c * . =
Solution,
a
varies $ointly as b and c % and a = () when
=
a = kbc () = k +(+* k =
substitute the set of 'iven data to find k
()
a!!ly the !ro!erties of e#uality
&'
k ( =
Therefore% the re#uired e#uation of variation is, a . z varies $ointly as x and y . If z &) when x variation and the e#uation of the relation. =
Solution, ,
=
=
(bc
* and y = ) % find the constant of
z = kxy &) = k +*+) k =
substitute the set of 'iven data to find k
&)
a!!ly the !ro!erties of e#uality
'* k =
' (
The e#uation of the variation is, z
=
' xy (
The area A of a trian'le varies $ointly as the base b and the altitude a of the trian'le. If A = )-cm' when b &.cm and a &(cm % find the area of a trian'le whose base is 2cm and altitude is &&cm.
.
=
Solution, A = kab ,
the e#uation of the relation
)- = k +&(+&. k =
=
substitute the set of 'iven data to find k
)-
a!!ly the !ro!erties of e#uality
&(. k =
& '
*
The e#uation of the variation is, A
& =
ab
'
Therefore% when a = && and b = / % the area of the trian'le is A
& =
+&&,+/,
' A
=
**cm '
. The area A of rectan'le varies $ointly as the len'th l and the width w and A = &/.cm' when l = 0cm and w = -cm . 8ind the area of a rectan'le whose len'th is '.cm and whose width is -cm . Solution , ,
A
=
&/.
=
k +0,+-,
k =
&/.
k
*
=
A A
substitute the set of 'iven data to find k a!!ly the !ro!erties of e#uality
*-
Therefore% when l A
the e#uation of the relation
klw
= 0cm and
=
*lw
=
*+'.,+-,
=
*..cm
w
=
-cm .
'
/. The volume ( V ) of a !rism on a s#uare base varies $ointly as the hei'ht ( h ) and the s#uare of a side ( s ) of the base of the !rism. If the volume is /&cm( when a side of the base is cm and the hei'ht is 0 cm% write the e#uation of the relation. Solution, E9!ress the relation as,
V
=
/&
=
ks ' h k +*, ' +),
substitute the 'iven values for V %s and h
-
/& = k +&)+) k
/& =
k =
reduce to lowest term
0)
'1 ('
The e#uation of variation is V 0.
=
'1 ' s h ('
E9tendin' the !roblem on the !revious e9am!le% find the volume of the !rism if a side of the base is 1 cm and the hei'ht is & cm. Solution,
V
=
'1 ' s h ('
from the !revious e9am!le
V
=
'1 +1, '+&', ('
substitute the 'iven values for s and h
V
=
'1 +*0,+&', ('
V
=
'1 +-//, ('
&-/1) (' V = *0)2&'-cm( V
1.
=
The volume ( V ) of a !rism on a s#uare base varies $ointly as the hei'ht ( h ) and the s#uare of a side ( s ) of the base of the !rism. A. If the volume is 1 cm when a side of the base is cm and the hei'ht is 2 cm% write the e#uation of the relation.
Solution, V
1'
=
=
'
ks h
k +(, ' +/,
where k is the constant of variation substitute the 'iven to find k
)
1'
:
=
k +0,+/,
1' = 1'k 1' k = 1' k = & The e#uation of the relation is V
= s 'h
. 8ind the volume when a side of the base is / cm and the hei'ht is & cm. Solution, V = +- ' +&*
substitute the 'iven values for s and h
V = +'-+&* V
= (-.cm(
C. y how many !ercent is the ori'inal volume V & increased if a side is increased by &4; and the hei'ht is 4;. Solution, &.&s since the side is increased by &4; and H > &.h since the hei'ht is increased by 4;. Then the new volume V ' is,
Since V&
V'
= S ' H
V'
= +&2&s ' +&2'h
V'
= +&2'&s ' +&2'h
V'
= &2*-'s 'h
= s 'h % we may substitute V & into the results% which 'ives V'
=
&2*-'V & .
The increase in volume is 4./ of the ori'inal volume V & . To chan'e 4./ to !ercenta'e% multi!ly it by &44; that will 'ive you,
+.2*-'+&..3 = *-2'3
1
The followin' illustrations are a!!lications of variation in different fields of mathematics li+e ?eometry% En'ineerin'% etc. Examples:
&. The volume of a ri'ht circular cylinder varies $ointly as the hei'ht and the s#uare of the radius. The volume of a ri'ht circular cylinder% with radius centimetres and hei'ht 1 centimetres% is / cm . 8ind the volume of another cylinder with radius 2 centimetres and hei'ht & centimetres. r Solution, The e#uation of the relation is V
=
khr '
8rom the 'iven set of data, r = * cm
h = 1 cm V
=
(-' cm(
To find + substitute the values above,
V
=
k = k = k = k =
khr ' V hr '
rearran'in' the e#uation above
(-' +1+*' (-' +1+&) '' 1
sim!lifyin' the fraction
To find the volume of a cylinder with r > 2 cm and h > & cm,
V
=
'' +&(,+/, ' 1
V
=
'' +&*,+)*, 1
V
=
'/&) cm( /
4
. The horse!ower h re#uired to !ro!el a shi! varies directly as the cube of its s!eed s . 8ind the ratio of the !ower re#uired at & +nots to that re#uired at 1 +nots. Solution, The e#uation of the relation is h = ks ( The ratio of !ower re#uired at & +nots to 1 +nots is
h' h& h' h& h' h& h' h&
= =
k +&*( k +1 ( +&*(
the k 5 s cancel out
+1 (
=
'1**
=
/
(*(
&
. The !ressure P on the bottom of a swimmin' !ool varies directly as the de!th d of the water. If the !ressure & is &/ @a when the water is metres dee!% find the !ressure when it is ./ metres dee!. Solution &,
P
=
kd
k =
P
k =
&'-
d '
solvin' for the constant of variation
since @ > &/ when d >
k = )'2 P
=
)'2-d
P
=
+)'2-,+*2-,
P
=
'/&2'- @a
&
"re##ure i# defined a# t4e force exerted per unit area "a#cal +"a i# t4e metric unit for pre##ure
'
0
Solution , In this solution% you do not need to find +. The e#uation P kd P P maybe written as k = % meanin' that the ratio is a constant. Therefore, d d P& P ' =
d&
=
&'-
d '
=
'
P ' *2-
+&'-+*2-
P '
=
P '
= '/&2'- @a
'
. The horse!ower re#uired to !ro!el a shi! varies directly as the cube of its s!eed. If the horse!ower re#uired for a s!eed of &/ +nots is &4 &/% find the horse!ower re#uired for a s!eed of 4 +nots. Solution, let @ > re#uired horse!ower s > s!eed% in +nots Since @ varies directly as s % you have
P = ks (
(&)
&.,&'- = k +&-( k
=
k
=
&.,&'+&- ( &.,&'(, (1-
k = ( P = (+'.(
substitute + > and s > 4 in (&)
P = (+/... P = '*, ... h!
&.
/. The wei'ht of a rectan'ular bloc+ of metal varies $ointly as its len'th% width and thic+ness. If the wei'ht of a & by 2 by 0 dm bloc+ of aluminum is &2.1 +'% find the wei'ht of a &0 by &4 by dm bloc+ of aluminum. Solution, 5et W = weight in +ilo'rams l = length in decimeters w = width in decimeters t = thickness in decimeters Since the wei'ht of the metal bloc+ varies $ointly as its len'th% width and thic+ness you have,
W
= klwt
k = k =
k =
W lwt &/21 +&(+/+) &/21 -1)
Substitute k =
&/21
% l &) % w &. and t * in the e#uation W -1) the wei'ht of the desired bloc+,
W =
&/21 -1)
=
=
=
= klwt to 'et
+&)+&.+*
W = '.2/ +'
wei'ht of the &0 by &4 by dm bloc+
0. The amount of coal used by a steamshi! travelin' at uniform s!eed varies $ointly as the distance traveled and the s#uare of the s!eed. If a steamshi! uses / tons of coal travelin' 24 +m at &/ +nots% how many tons will it use if it travels &4 +m at 4 +nots Solution, 5et T > number of tons used s > the distance in miles v > the s!eed in +nots and then T > +(sv ) (&) hence% when T > /% s > 24 and v > &/% you have &&
/ > +(24)(&/ ) + > BBB/BBB (24)(/) + > BB&BB 44 Substitutin' this value for + in (&)% you have T > BB&BB (&4)(4 ) 44 T > 2444 44 T > &4 tons Try this out A. Translate each statement into mathematical statement. *se k as the constant of variation. &.
P varies $ointly as q and r .
.
V varies $ointly with l % w and h .
.
The area A of a !arallelo'ram varies $ointly as the base b and altitude h .
.
The volume of a cylinder V varies $ointly as its hei'ht h and the s#uare of the radius r .
/.
The heat H !roduced by an electric lam! varies $ointly as the resistance R and the s#uare of the current c .
0.
The area A of a !arallelo'ram varies $ointly as the base b and altitude a
1. The volume V of a !yramid varies $ointly as the base area b and the altitude a . 2. The area A of a trian'le varies $ointly as onehalf the base b and the altitude h 2 3. The a!!ro!riate len'th ( s) of a rectan'ular beam varies $ointly as its width ( w ) and its de!th (d ). &4. The area A of a s#uare varies $ointly as its dia'onals d & and d ' .
&'
. Solve for the value of the constant of variation k % then find the missin' value. &. D varies $ointly as 9 and y and D > 0 when 9 > / and y > 0. a. find D when 9 > 1 and y > 2 b. find 9 when D > 1 and y > c. find y when D > 2 and 9 > . z varies $ointly as x and y . If z ( when x and y = 0 . =
=
( and y = &- % find z when x
=
)
(. z varies $ointly as the s#uare root of the !roduct x and y . If z = ( when x = ( and y = &' % find x when z = ) and y = / .
. d varies $ointly as o and '. If d > &/% when o > & and ' > /% find ' when o > & and d > 2. /. # varies $ointly as r and s. If # > .% when r > 4.0 and s > 4.2% find # when r > &.0 and s > .4&. 0. d varies $ointly as e and l . If d > .% when e > 4.0 and l > 4.2% find d when e > &.0 and l > .4&. 1. 9 varies $ointly as w% y and D. If 9 > &2% when w > % y > 0 and D > /% find 9 when w > /% y > & and D > . 2. D varies $ointly as 9 and y. D > 04 when 9 > and y > . 8ind y when D >24 and 9 > . 3. The wei'ht W of a cylindrical metal varies $ointly as its len'th l and the its diameter d
s#uare of
a. If W > 0 +' when l > 0 cm and d > cm% find the e#uation of variation. b. 8ind l when W > &4 +' and d > cm. c. 8ind W when d > 0 cm and l > &. cm. &4. The amount of 'asoline used by a car varies $ointly as the distance traveled and the s#uare root of the s!eed. Su!!ose a car used / liters on a &44 +ilometer tri! at &44 +mhr. About how many liters will it use on a &3 +ilometer tri! at 0 +mhr
C. 7hat did the !i' say when the man 'rabbed him by the tail &(
-
-.
&)
(.
*
* */
/
(
-
(
(
(
'
'
&0'
'&
*
*.
&' &/
I If D varies $ointly as 9 and y% and D > %
S F varies $ointly as 9 and y and D > 04
when 9 > and y > % find D when 9 > and y > /
when 9 > and y > . 8ind y when D > 24 and 9 > .
G If D varies $ointly as 9 and y and D > &% E If w varies $ointly as 9 and y and w > 0 when 9 > and y > % find the constant of variation.
when 9 > and y > % find the constant of variation.
S If D varies $ointly as 9 and y and D > % T If A varies $ointly as l and w and A is 0 when 9 > and y > % find D when 9 > and y > .
when l > 3 and w > % find A if l > 0 and w > .
H If a varies $ointly as c and d % and
6 If w varies $ointly as 9 and y and
a > 4% when c > and d > . 8ind d when a > / and c > 2.
w > when 9 > and y > % find the value of w when 9 > 3 and y > .
E If D varies $ointly as 9 and the s#uare of H If varies $ointly as w and l and A > 2
y and D > 4% when 9 > and y > . 8ind D when 9 > and y > .
when w > and l > % find the value of A when w > 3 and l > &/.
T If D varies $ointly as 9 and the s#uare of M 9 varies $ointly as and y and D > 2% y and D > 4% when 9 > / and y > . 8ind D when 9 > and y > /.
when 9 > and y > % find the constant of variation.
8 If y varies directly as 9 and if y > &/
I ! varies $ointly as r and s and ! >
when 9 > /% find the value of y if 9 > 1.
when r > and s > . 8ind the constant of variation.
E If w varies $ointly as 9 and y and if w > < If y varies directly as 9 and y > 0 when &/ when 9 > and y > % find the value of w if 9 > and y > .
9 > 2% what is the value of y when 9 >
&*
5esson Combined variation Combined variation is another !hysical relationshi! amon' variables. This is the +ind of variation that involves both the direct and inverse variations. This relationshi! amon' variables will be well illustrated in the followin' e9am!les. Examples:
A. The followin' are mathematical statements that show combined variations. &. k = . k =
I Prt E IR
. k = .
Pv t
/. k =
c ar
= k ab ' c
. Translate each statement into a mathematical statement. *se + as the constant of variation. &. T varies directly as a and inversely as b.
T =
. varies directly as 9 and inversely as the s#uare of D.
=
. @ varies directly as the s#uare of 9 and inversely as s. . The time t re#uired to travel is directly !ro!ortional to the tem!erature T and inversely !ro!ortional to the !ressure @. &-
P =
t =
ak b kx z ' kx' s kT P
P =
/. The !ressure @ of a 'as varies directly as its tem!erature t and inversely as its volume -.
kt V
The followin' e9am!les are combined variation where some terms are un+nown and can be obtained by the available information. C. If D varies directly as 9 and inversely as y% and D > 3 when 9 > 0 and y > % find D when 9 > 2 and y > &. Solution, The e#uation is z =
kx y
Substitutin' the 'iven values,
0=
k )
k =
0
' (
k = ( z =
+(+/ &'
z = ' <. x varies directly as y and inversely as z . If x when y = &' and z '. .
=
&- when y = '. and z *. % find x =
=
Solution, The e#uation is x =
ky z
Substitutin' the 'iven values to find + where x
&- = k =
k '. *.
+&-+*. '.
k = (. &)
=
&- when y = '. and z *. % =
To find x when y = &' and z
ky
*sin' the e#uation x =
x =
'.
=
z +(.+&' '.
x = &/ E. t varies directly as m and inversely as the s#uare of n . if t &) when m n = ' % find t when m = &( and n = ( . =
=
/ and
Solution, The e#uation of the variation, t = To find k % where t &) % m =
&) = k = k = k =
=
t = t =
=
k +/ +' '
&)+'' / +&)+* / )*
&( and n
=
+/+&( +(' &.* 0
t = &&
n'
/ and n = ' % substitute the 'iven values
/ k = / To find t when m
km
or
0
&1
(
8.. r varies $ointly as s and t and inversely as ! . If r =
! = -) % find r when s = ) % t = 1 and ! = /* .
( '/
when s
=
&. % t ( and =
Solution, The e#uation of the variation, r =
kst !
Substitute the 'iven values to find k , ( '/ k = k = k =
=
k +&.+( -) +(+-)
+'/+&.+( ' &. & -
To find r when s = ) % t = 1 and ! = /* . r =
kst !
& +)+1 r = /* *' & r = - /* & r = &. ?. ?iven, w varies directly as the !roduct of x and y and inversely as the s#uare of z . If w = 0 when x = ) % y = '1 and z = ( % find w when x = * % y = 1 and z = ' . Solution, The e#uation, w =
kxy
z ' Substitutin' the first 'iven set of values to the e#uation% where w = 0 % x = ) % y = '1 and z ( =
&/
0=
k +)+'1
0=
k +&)'
+('
0
/& = &)'k k = k =
/&
or
&)' & '
8ind the value of w when k = and z = ' % you have w=
& '
and use the second set of values when x
=
* % y = 1
kxy z '
&
+*+1
w =
'
w =
+'+1
w =
1
'' * or
'
w = (2H. The current I varies directly as the electromotive force E and inversely as the resistance R . If in a system a current of 4 A flows throu'h a resistance of 4 Ω with an electromotive force of &44 -% find the current that &/4 - will send throu'h the system. Solution, 5et
I > the current in A (am!ere) E > electromotive force in - (volts) R > Ω ( ohms)
The e#uation, I =
kE R
Substitute the first set of 'iven data,
&0
I > 4 A E > &44 R > 4 Ω
y substitution% find +, k &.. '. = '. k =
+'.+'.
k =
*..
&.. &..
k * =
To find how much (I) current that &/4 - will send throu'h the system I =
+*+&-.
I = (.
'.
Gotice% the system offers a resistance of 4
Ω.
Try this out A. *sin' k as the constant of variation% write the e#uation of variation for each of the followin'. &. W varies $ointly as the s#uare of a and c and inversely as b. . The electrical resistance ( R ) of a wire varies directly as its len'th ( l ) and inversely as the s#uare of its diameter ( d ). . The acceleration A of a movin' ob$ect varies directly as the distance d it travels and varies inversely as the s#uare of the time t it travels. . The heat H !roduced by an electric lam! varies $ointly as the resistance R and the s#uare of the current C . /. The +inetic ener'y E of a movin' ob$ect varies $ointly as the mass m of the ob$ect and the s#uare of the velocity v . . Solve the followin' &. If r varies directly as s and inversely as the s#uare of u% then r > when '.
s > &2 and u > . 8ind,
a. r when u > and s > 1. b. s when u > and r > c. u when r > & and s > 0 . p varies directly as q and the s#uare of r and inversely as s. a. b. c. d.
write the e#uation of the relation find k when p > 4% q > /% r > and s > 0 find p when q > 2% r > 0 and s > 3 find s when p > &4% q > / and r > .
. w varies directly as xy and inversely as v and w > &44 when x > % y > 3 and v > 0. 8ind w when x > % y > & and v > 3. ( . Su!!ose " varies directly as b ' and inversely as s( . If " = when b * s ' % find b when " = ) and s * . =
=
Let ‘s summarize
'&
=
) and
What have you learned A The followin' are formulas and e#uations which are fre#uently used in mathematics and in science. State whether the relationshi! is considered direct% inverse% $oint or combined variation.
BBBBBBBBBBBBBBBBBBBB&. C > Π BBBBBBBBBBBBBBBBBBBB. A > lw BBBBBBBBBBBBBBBBBBBB. < > rt BBBBBBBBBBBBBBBBBBBB. I > !rt BBBBBBBBBBBBBBBBBBBB/. - > lwh BBBBBBBBBBBBBBBBBBBB0. A > Πr BBBBBBBBBBBBBBBBBBBB1. - > Πr BBBBBBBBBBBBBBBBBBBB2. E > mc BBBBBBBBBBBBBBBBBBBB3. 8 > ma BBBBBBBBBBBBBBBBBBB&4. - > Πr T & ' BBBBBBBBBBBBBBBBBBBBB&&. > mv ' BBBBBBBBBBBBBBBBBBBBB&. @ > 8 A BBBBBBBBBBBBBBBBBBBBB&. 7 > 8d BBBBBBBBBBBBBBBBBBBBB&. - > IJ BBBBBBBBBBBBBBBBBBBBB&/. K > mc δt . 8or each 'iven e#uation with + as the constant of variation and solve for the un+nown value. Choose the letter of the correct value of the un+nown. &. w varies inversely as D and w is when D is 0. what is D when w is & a. &1 b. &&1 c. &41 d. none of these . 9 varies $ointly as y and D. 9 is when y is and D is . 7hat is D if 9 is 2 and y is &4 a. /0 b. 0/ c. 0 d. / . m varies directly as n but inversely as !. 7hat is n if m is &0 and ! is &2 a. 2 b. 20 c. 22 d. 34 . ! varies inversely as # and r% and ! > when # is and r is &. 7hat is # when ! is 0 and r is &4 a. /2 b. 4/ c. / d. 2/ /. 8 varies directly as ' and inversely as the s#uare root of the !roduct of I and h% and ''
8 > / when ' > 1./% I > and h > &2. 7hat is 8 when ' > % I > L and h > &0 a. 2 b. &0 c. &2 d. none of these 0. 7 varies directly as u and v% and 7 is 1/ when u is / and v is 3. 7hat is v when 7 is &/4 and u is &4 a. b. ./ c. / d. none of these 1. S varies directly as t and inversely as u % and S is 3 when t is and u is &. 7hat is t when u is 2 and S is &0 a. .& b. &.&0 c. .&0 d. none of these 2. y varies directly as w and inversely as the cube of 9. 7hat is y when w is and 9 is a. 23 b. 3 c. 2 d. 32 3. @ varies as the !roduct r and inversely as the s#uare of t. 7hat is t when @ > % r > a.
(( '
b.
c. &&
d. none of these
&4. y varies directly as 9% and y > 0 when 9 > . 7hat is 9 if y > & a. b. & c. d. 4
'(
Answer ey How much do you +now A. &. A > +ba or A > +ab
- > +e
. - > +ba or - > +ab
1. 8 > +w
. A > +r . # = .
2. h >
km' n
3. m =
k r ' kr
s &4. K > +JT
- > +lwh
Try this out 5esson & A. &. @ > +#r
0. A > +ab
. - > +lwh
1. - > +ab
. A > +bh
&
2. A >
'
. - > +hr
bh
3. s > +wd
/. H > krc '
&4. A > +d &d
B2 &2 $
= kxy , k = (&
)2 k - , d .2./ =
&-
a2 &&-21 62 /21& c2 020' '2 k =
& &-
( &*
12 x = kwy , k =
( &.
or .2(, x
/2 z = kxy , k - , y = / =
, z = (2)
(2 z = k xy , k = *2 k =
=
& '
02 a2 k =
&
,W
& =
0 0 62 l = ''2- cm c2 -2) 7$
, x = &/
, g = &2/
ld '
&.2 k .2.'- , (/2* liter# =
-2 k = -. , q = .2/
'*
=
-*
C. 7hat did the !i' say when the man 'rabbed him by the tail T -.
8 -9*
I (.
T */
8 *9(
E (
O &0'
F '&
I /
S &'
E -9'
N (9'
D &/
*
E *.
S &)9(
5esson A. &. W
ka 'c =
b kl
. R =
d '
. A > +d t
= kR% '
. H
/. E > +mv . &. r =
ks
% k =
. + > &44 & w = &(( (
*
!' 0 a. r > b. s > 0 d. u >
. a. P =
. k =
&
) b > 2
'
kqr s
b. + > c. @ > 30 d. s > 0
'-
7hat have you learned A.
. &. direct . $oint . $oint . $oint /. $oint 0. direct 1. direct 2. $oint 3. $oint &4. combined &&. $oint &. combined &. $oint &. $oint &/. $oint
E9am!le /.
&. a . b .c . d /. a 0. b 1. c 2. d 3. a &4. b
If the volume of a mass of 'as at 'iven tem!erature is /0 in when the !ressure is &2 lbin % use oyleNs law to find the volume when the !ressure is &0 lbin
Solution, oyleNs law states that - > +@ or @- > +% meanin' that @- is a constant. a) without findin' +% you may write @&-& > @- where @& > &2 lbin% - & >% /0 in% @> &0lbin% - > Substitutin' (/0)(&2) > (&0) - - > /0(&2) 0 - > 0 in
E9am!le 2. The load which can be safely !ut on a beam with a rectan'ular cross section that is su!!orted at each end varies $ointly as the !roduct of the width and the s#uare of the de!th and inversely as the len'th of the beam between su!!orts. If the safe load of a beam in wide and 0 in dee! with su!!orts 2 ft a!art is 144 lb%
')
find the safe load of a beam of the same material that is in wide and &4 dee! with su!!orts & ft a!art.
Solution, let
w > width of beam% in inches d > de!th of beam% in inches l > len'th between su!!orts% in feet 5 > safe load% in !ounds
5 > +wd l Accordin' to the first set of data %when w > % d> 0 and l > 2% then 5 > 144% therefore then
144 > + () (0 ) 2
+ > 2(144) > BB2(144)BB (0) &42 + > 44 Conse#uently% if w > % d > &4% l> & and + > 44% you have
5 > 44()(&4 ) &
5 > 0000
'1
