Solid mensuration

Neil Neil Bryan C. Bryan C. Hernandez ECE 5101 Hernandez ECE 5101 Solid Mensuration Solid Mensuration 1.

1. The radius The radius of a sof a sphere is phere is 5 feet. 5 feet. FFind the arind the area of tea of the grhe great ireat irle of thle of this sphere.is sphere. a. a. 55   b. 25 b. 25   .. 11!!55   d. "!5 d. "!5   !.

!. # yli# ylinder ander and a ond a one eane eah ha$e a h ha$e a radiradius of % us of % in. anin. and a hed a height oight of & in. f & in. 'hat'hat is the ratio of the

is the ratio of the

$olu(e of the one to the

$olu(e of the one to the $olu(e of the ylinder)$olu(e of the ylinder) a.

a. **oneone+ *+ *ylinderylinder , 1 + !, 1 + !

b.

b. VVconecone: V: Vcylindercylinder = 1 : 3= 1 : 3 .

. **oneone + *+ *ylinderylinder, 1 +-, 1

+-d.

d. **oneone + *+ *ylinderylinder, 1 + 1, 1 + 1

%.

%. FFind the $oluind the $olu(e of a pris( hos(e of a pris( hose /ase is an euilae /ase is an euilaterateral triangl triangle ith a side of le ith a side of   and

 and hose length hose length is ".is ". a a.. !! /. /. ""22%% c c.. 22442233 d. & d. & .

. The suThe surfarfae are area of a rea of a regulaegular suarr suare pyrae pyra(id is 3 , 3 4(id is 3 , 3 4 1133 here33 here BB is theis the area of the /ase

area of the /ase p p isis

! !

the peri(eter of the /ase and

the peri(eter of the /ase and ss is the slant height. is the slant height. 6f the /ase is a suar6f the /ase is a suare ithe ith a side length of 10

a side length of 10 and the slant height is

and the slant height is 15 7nd the surfae area of the pyra(id.15 7nd the surfae area of the pyra(id. a

a.. &&55 / /.. !!!! .. %%11 d d.. 4400 5.

5. 'hih of the fo'hih of the folloinlloing for(g for(ulas ill 7nd the surulas ill 7nd the surfafae area of a yle area of a ylinder itinder ithh one open end and one losed end)

one open end and one losed end)

a. a. S# , !-rh 4 !-rS# , !-rh 4 !-r!! b. b. SA SA = = 2πrh 2πrh + + πrπr22 . . S# , !-rh 8 -rS# , !-rh 8 -r!! d d.. SS# # , , !!--rrhh ".

". 'hih pl'hih platonatoni solid i solid ill /e rill /e reateeated hen the ned hen the net shot shon at the right isn at the right is asse(/led)

asse(/led) a

a.. TTeetrtraaheheddrroonn /

/.. CCuu//ee

c.

c. OcOctatahehedrdronon

d. dodeahedron d. dodeahedron

9.

9. Soda is soSoda is sold in alu(inld in alu(inu( ans thau( ans that (east (easure " inhure " inhes in height aes in height and ! inhend ! inhess in dia(eter

in dia(eter. . Ho Ho (any u/i inhes of so(any u/i inhes of soda are ontada are ontained in a full an)ined in a full an) :;ou

:;ound anser to nd anser to thethe nearest tenthnearest tenth of a of a u/i u/i inh.<)inh.<) a a.. 11!!.. b b.. 1188.. .. !!.. d d.. 9955.. &.

&. FFind ind the the $o$olu(lu(e te to tho thee nearest cubic inchnearest cubic inch of a one hose radius is 1! of a one hose radius is 1! inhes and

inhes and hose height hose height is 15 is 15 inhes.inhes. a

a.. !!&&!!

b

b.. 222266

.. 55""55 d

d.. 11&&&& =.

=. TTop soil op soil is sold is sold /y the /y the u/i yu/i yard. ard. TTo theo the nearest cubic yardnearest cubic yard ho (uh top soil ho (uh top soil is needed to reate

is needed to reate a >oer /ed ith soil  feet ide /y !5 feet long /y "a >oer /ed ith soil  feet ide /y !5 feet long /y " inhes deep) inhes deep) a a.. 5500 / /.. !!99 . . "" d. d. 22 10.

10.o! "anyo! "any sides does a polygon ha$e if the su( of sides does a polygon ha$e if the su( of its interior angles is 9!0?)its interior angles is 9!0?) a. a. 55 b. b. 66 . . 99 d. d. &&

11.Find the nu(/er of degree

11.Find the nu(/er of degrees in eah s in eah interior angle of ainterior angle of a dodeagon) dodeagon) a. a. 1%5?1%5? /. 1? /. 1? c c.. 115500## d.

d. Cannot Cannot /e /e deter(ineddeter(ined 1!.Find the su( of the

1!.Find the su( of the e@terior angles of a 15 e@terior angles of a 15 sided polygon.sided polygon. a a.. !!%% b b.. 336600 .. 1155"" d d.. !! 1%.6f to planes a

1%.6f to planes are perpendire perpendiular to the ular to the sa(e linesa(e line a.

a. ThThey aey arre pee perprpenendidiuulalarr..

b.

b. $hey $hey are are %arallel.%arallel.

.

. They They interinterset set /ut a/ut are nre not pot perpeerpendiundiularlar.. d.

d. NoNone ne of of ththe ae a/o/o$e$e.. 1.'hat i

1.'hat is a polygon alled if the su( s a polygon alled if the su( of its interior angles euals 10?)of its interior angles euals 10?) a a.. HHee@@aaggoonn / /.. AAttaaggoonn c c.. &&eeccaa''oonn d. odeagon d. odeagon

15.Eah interior angle of a regular polygon (easures 1"!

15.Eah interior angle of a regular polygon (easures 1"!?. ?. Ho (any sides doesHo (any sides does the polygon ha$e)

the polygon ha$e)

a a.. 2200 / /.. 11&& .. 11"" d. d. CaCannnnoot t //ee

1".'hen asse(/led the net shon at the right ill reate a+ 1".'hen asse(/led the net shon at the right ill reate a+

a a.. CCuu//ee b. (ylinder b. (ylinder .. CCoonnee d d.. rriiss((

19.Find the $olu(e of a he(isphere hose radius is 10 feet. ;ound anser to the 19.Find the $olu(e of a he(isphere hose radius is 10 feet. ;ound anser to the

nearest cubic foot. nearest cubic foot. a a.. 11== / /.. 1100 c c.. 2200)) d d.. 11&&

1&.Ho (any sides in a u/e) 1&.Ho (any sides in a u/e)

a. a.  b. b. 66 . . == d d.. 11!!

1=.Find the (easure of eah e@terior angle of 1=.Find the (easure of eah e@terior angle of aa

rregegulular ar hehe@a@agogon. a.n. a. =0=0?? /. 9!? /. 9!? c c.. 6600## d. 5? d. 5? !0.;egard

!0.;egarding the diagra( at the right hiing the diagra( at the right hih of the folloing state(ents is T;DE)h of the folloing state(ents is T;DE) a

a.. llaanneess ll andand qq are parallel planes.are parallel planes. /

/.. llaanneess ll and  interset in line E.and  interset in line E. c

c.. **ooiinntt PP _{i in %lanei in %lane} l.l.

d.

d. None None of of the the state(ents state(ents are are true.true.

!1.The legs of a right triangle (easure & and 15.

!1.The legs of a right triangle (easure & and 15. 6f a leg and hypotenuse of6f a leg and hypotenuse of another right tr

another right triangle are iangle are & and 19 r& and 19 respeti$ely these tespeti$ely these to right triangles aro right triangles aree ongruent. ongruent. a. a. $,-$,- /. F#SE /. F#SE !!.6n triangle

!!.6n triangle  ABC ABC <A<A is o/is o/tustuse.e. 'hi'hih stah state(te(ent ent is tis true arue a/ou/out tht thee su( of the (easures of

su( of the (easures of <B<Bandand <C<C)) a. a. ((<B +<B + ((<C<C , , =0=0 /. /. ((<B +<B + ((<C<C   =0=0 . . ((<B +<B + ((<C<C G G =0=0 d. d. ""<B +<B + _""<C<C _{= 180= 180}

!%.'hih of

!%.'hih of the folloing lengths are the sides of a the folloing lengths are the sides of a right triangle)right triangle)

a. a. 2.2.4/ 4/ 3.3.2/ 42/ 4 /. /. ..% % ..  .5.5 .. %.%.11  .!.!  .&.& d. d. ##ll ll oof tf thhee

!.#t a ertain ti(e of the day the shado of a 5 /oy is & long.

!.#t a ertain ti(e of the day the shado of a 5 /oy is & long. The shado of The shado of  a near/y >agp

a near/y >agpole at ole at this sa(e ti(e is !& longthis sa(e ti(e is !& long. . Ho tall is the >agpole)Ho tall is the >agpole) a. &.5 a. &.5 /. 1" /. 1" c c.. 11..55 d. !0 d. !0 !5.6n triangle

!5.6n triangle ABC, m<A ABC, m<A, "0 and, "0 and m<Bm<B , 0. , 0. 'hih side o'hih side of trianglef triangle ABC ABCis theis the longest) longest) a a.. ##CC b. A b. A .. BBCC d d.. BBCC##BB !".# fris/ee lands on

!".# fris/ee lands on the top of a 15 onrete all. the top of a 15 onrete all. TTo retrie$e the fris/ee ao retrie$e the fris/ee a ladder (ust /e pla

ladder (ust /e plaed ed suh that the foot of thsuh that the foot of the ladder is " feet froe ladder is " feet fro( the /ase( the /ase of the all and

of the all and the top of the top of the ladder rthe ladder rests ests on the top on the top of the all. of the all. 'hih'hih hoie is the shortest length ladder that an /e used)

hoie is the shortest length ladder that an /e used) a. 15 a. 15 /. 1" /. 1" c c.. 11 d. 1&I d. 1&I

!9.The sides of a triangle are 5 " and 10. Find the length of the longest side of  !9.The sides of a triangle are 5 " and 10. Find the length of the longest side of 

a si(ilar triangle

a si(ilar triangle hose shorthose shortest side est side is 15.is 15. a a.. 1100 / /.. 1155 .. 11&& d. d. 33

!&.The $erte@ angle of an

!&.The $erte@ angle of an isoseles triangle (easures eight ti(es the (easure of isoseles triangle (easures eight ti(es the (easure of  a /a

a /ase angse angle. le. FFind ind the (easurthe (easure of e of a /a /ase aase angle.ngle. a. 1&? a. 1&? /. !? /. !? .. %%""?? d. %? d. %? !=.6n the

!=.6n the sa(e irle or ongruent irles ongruent sa(e irle or ongruent irles ongruent entral angles ha$eentral angles ha$e ongruent ars. ongruent ars. a. a. $,-$,- /. F#SE /. F#SE

%0.Find the (idpoint of the seg(ent Joining the

%0.Find the (idpoint of the seg(ent Joining the points : 8!< and points : 8!< and :8&"<.:8&"<. a a.. ::""  << / /.. ::88""88 .. ::!!  !!<< d d.. 22//

%1.Ho (any suare feet of tin are reuired to (aKe a funnel if the

dia(eters of the top and /otto( are !& in. and 1 in. respeti$ely and the height is ! in.) 3 , 1 :!&< , 1  . ! 3 , 1:1< , 9  . ! L , !  .  ! , :3  3<! 4 L!  ! , :1  9<! 4 !!   , !5  . 3 _ ,  :3 4 3<  ,  :1 4 9<:!5< 3 _ , 5!5  .! 3 _ , 5!5  . ! _: 1 33. ! < 1!  .   _, 33. 333 33.3

%!.# stone /ridge has three eual se(iirular arhes resting on four eual_{retangular piers. The} radius of eah arh is ; the di(ensions of the piers  ' H hile the

distane fro( the top of eah pier to the top of the /ridge is . 6f  , 50 ft. ' , 5 ft. H , 10 ft. ; , " ft.  , 1! ft. 7nd the nu(/er of u/i yards of (aterial used in /uilding the /ridge.

*olu(e , * , #/ 'here 3 , !!:5"<  % 1 :"! ! 3₃ , 90!.%5 33.! <O  %1!:10<O  Thus 3 , 90!.%5:50< , %5119.9 33.% 3 , 3333. 3 33.3

%%.The great Chinese all is said to /e 1500 (iles long !0 ft. high 15 ft ide at the top and !5 ft.

ide at the /otto(. 6f it ere possi/le to /uild ith this (aterial a all around the earth at the euator of a unifor( thiKness of  ft. ho high ould it /e (ade) :The euator is appro@i(ately !=00 (iles long.<

Chinese all *,#/ #/,1!:!5415 <:!0< #/,00 ft.! ,1500:5!&0< ,9=!0000ft. 3

*,00:9=!0000 < *,%1"&000000 ft.% Euatorial all *,#/ %1"&000000,h:!=00P5!&0<%1"&000000,h:!=00P5!&0< h=6.0241 7t.

%.# hole " in. in dia(eter as /ored through a sphere 10 in. in dia(eter. Find the $olu(e of the part ut out.

By %885 triangle @ , in.

h1 , !@ , & in.

h! , Q :10 R h1< , 1 in. *olu(e ut out

3 , 3_{3 3 3 333}4 3_3333333 3 ,  :%!<:L₁< 4 !   L_! ! % %:5<  L_!O 3 ,  :%!<:L₁< 4 !   !_%1 %:5<  1O 3 , 333  33.3, 333. 33 333 3

%5.'hen a liuid freezes it hanges into a (ass of solid /odies eah of hih has a de7nite

geo(etri for( and is Knon as a rystal. Crystals are alays /ounded /y plane surfaes hih are arranged in an orderly fashion ith referene to i(aginary lines dran through the rystal and alled its area. E$ery rystal has therefore a de7nite geo(etri for(. Co(pare the $olu(es of rystals one in the shape of a he@agonal right pris( a units on a side and !a units

in altitude and the other in the shape of to oppositely direted he@agonal regular pyra(ids eah a units high ith a o((on /ase a units on a side. 3 , " 3 1 3! ! sin "03 3₃ , % % 3 ! 3�3!

Crystal pris( :h1 , !a units<

3₁ , 3₃L₁ , % % 3 ! :!3< 3₁ , %2% 3% 333%

Crystal pyra(id :h! , a units<

 1 1 %2% _! 3_! , ! 3 % 33L! , ! 3 % : ! 3 <:3<  33_, 3 23 33 3_{3 23 3}3

$h/ the 9ol"e o7 crytal %ri" i three ti"e the 9ol"e o7 the crytal %yra"id.

%".isregarding uality and onsidering oranges as spheres deter(ine hih i s the /etter /argain oranges !8% in. in dia(eter at 15 ents per dozen or oranges a$eraging %81! in. in dia(eter at %0 ents per dozen.

3

2 !

For oranges of !8% in. in dia(eter :radius , 1.%95 in.<  _% 1%%1 _% *olu(e 3₁ , 1! 3 %  :1.%95 < ,    . %! Dnit ost 3 , 15 33333 %!    .% , 0.1& 3333 3%

For oranges of %81! in. in dia(eter :radius , 1.95 in.%_<

 _% %% _% *olu(e 3_! , 1! 3 _%  :1.95 < ,    .  Dnit ost 33 33333 3333 3₃ , 333 , 3. 3333 3 333 3  33

$h/ the 30 cent %er doen i the better bar'ain.

%9.Ho (any u/i feet are there in the largest piee of ti(/er of suare ross setion that an /e ut fro( the log)

a! 4 a! , %! !a! , = a! , .5 /! 4 /! , !! ! /! ,  /! , ! 1

3₁, 3! , .5 33.!3_!, 3! , ! 33.! 3 , 1:3

4 3 4 23 3 <L

3 , 1:.5 4 42.5  !<:1&<

%

3 , 33 333

%&.6n the foundation orK of the 'oolorth Building a 558story /uilding in Ne  UorK City it as

neessary in order to reah the /edroK to penetrate the sand and uiKsand to a depth in so(e instanes of 1%1 ft. 6f the largest irular aisson 1= ft. in dia(eter as 1%0 ft. deep and as 7lled ith onrete to ithin %0 ft of the surfae ho (any u/i yards of onrete ere reuired) Conrete dia(eter , 1= ft. , 1=% yd.

epth of onrete , :1%0 8 %0<  % , 100% yd. *olu(e of onrete

1 1= ! 100

3 , 3₃L ,

  : % < : % <

3 , 3333. 33 33.3

%=.The inside of a $ase is an in$erted one !.=&% in. aross the top and 5.01" in. deep. 6f a hea$y

sphere !.=& in. in dia(eter is dropped into it hen the $ase is full of ater ho (uh ater ill o$er>o)

tan  , 1 .  =1 5 5.01" tan  , 1".5"? 3 , 3 sin , 1 . ! = sin 1".5"?

3 , .%&!   .

5.01"  L , 3  3 5.01"  h , .%&!  1.!= L , 1.&&%  . #(ount of ater that

o$er>os * , seg(ent of a sphere 3 ,L< , %1 L!:%3  1   :1.&&%!<%:1.!=<  1.&&%O % 3 , 3. 33 33.3

0.# stone is dropped into a irular tu/ 0 in. in dia(eter ausing the ater therein to rise !0 in.

'hat is the $olu(e of the stone)

 The $olu(e of the stone is eual to the $olu(e of ater that rises in the tu/ 3 , 3 L , 1  :0!  3 , &000  .% <:!0< 3 , &000  .% : 1 33.< 1!  . 3 , 33. 333 33.3

1.# fatory hi(ney has the shape of a frustu( of a regular pyra(id. 6ts height is 1&0 ft. and its upper and loer /ases are suares hose sides are 10 ft. and 1" ft. respeti$ely. The ross setion of the >ue is a suare hose side is 9 ft. Ho (any u/i feet of (aterial does the hi(ney ontain)

3 , 1:3 4 3 4 23 3 <L % 1 _{! ! ! !} 33 , _% 1" 4 10 4 21" :1 0 <O :1&0< 3₃ , %0="0 33.%

*olu(e of >ue :parallelepiped<

3₃ , 3₃L , 9! :1&0<

3

%

3₃ , &&!0 33.%

*olu(e of hi(ney

3 , 3₃ 4 3₃ , %0="0  &&!0

3 , 33 333 33.3

!.# druggist sells a ertain Kind of poder in a retangular /o@  /y ! /y 1 in. for !5 ents and in a ylindrial an !Q in. high and !Q in. in dia(eter for 10 ents. 'hih is the (ore eono(ial /uy)

6n retangular /o@ 3 , :!.5<:1.!5< , 1!.5  .% Dnit prie 6n ylindrial an !5 3 , 1!.5 1 33333 , !  .% Dnit prie 3 ,  :!.5 !_{<:!.5< ,} 1!.!91&"%  .% 10 3 , 1!.!91&"%, 0.&15 33333  .%

$he cylindrical can i "ore econo"ical.

%.#ssu(ing that a ity has 00 (iles of ater pipes and that the a$erage dia(eter of the pies is 1 ft. ho (uh ater is reuired to 7ll this entire syste() 3 , 3₃  3 , 1  :1!<:00 3 5!&0< , 5!&000 33.%  3 , 333 333 33.3 3 3. 33 333. , 33 333 333. 33 333. 3 33.3

.The ater supply of a distrit onsists of a laKe hose surfae area is 1000 ares. The nu(/er of household supplied ith ater is "0000. Supposing eah household to use !5 gal. per day alulate ho (any daysI supply :to the nearest day /elo the e@at nu(/er< is represented /y a depth of 1 ft. of ater in the laKe. :1 are euals %5"0 s. ft.<

aily ater onsu(ption of "0000 households

3₃ , !5:"0000< , 1500000 333. , !005%5 33.%

*olu(e of 1 ft. epth of ater in the reser$oir 3₃, 3₃L 3₃ , 1000 3 %5"0000 33.! L , 1 33. 3₃, %5"0000:1< , %5"0000 33.% Nu(/er of days ₃ 3    , 3₃ 33_{ 333}  333 , 333 333 , 333 3333

5.# ater tanK open at the top onsists of a right irular ylinder and a right irular one as

shon. 6f the altitude of the ylinder is three ti(es its radius and the altitude of the one is to ti(es the sa(e radius 7nd the nu(/er of suare feet of sheet (etal reuired to onstrut a tanK ha$ing a apaity of 10000 gal. :Ane gal. , !%1 u. in.<

3 , 10000 333 3 !%1 .% 1 33. % 3 : < 1 333. 3 , 1%%".&0" 33.% 3_{3 33333} 4 3₃₃₃₃ , 3 1!  .  3!L 4 1 3!L , 1%%".&0" % ! 11 _3%_{, 1%%".&0"} % 3 , .&9& 33.

L₁ , %:.&9&< , 1."% 33. L_! , !:.&9&< , =.95" 33.

3 , 3 _{ �M 3 33 33} 4 3 _M3333

3 , 2 L₁4 �

3 , 2:.&9&<:1."%< 4 :.&9&<  2.&9&! 4 =.95"!

  , 333. 33 33.3

".Ane hundred and 7fty posts are used in fening a lan. Eah post is /uilt in the for( of frustu(

of a pyra(id sur(ounted /y a pyra(id hose loer /ase is o((on ith the upper /ase of the frustu(. The height of the pyra(idal top is ! in. and the o((on /ase is a suare  in. on an edge. The loer /ase of the frustu( has an edge of " in. 6f the o$erall height of eah is " ft. ho (uh onrete ill /e used in (aKing the posts)

*olu(e of pyra(idal top _{1 1 } ! _! 3₃ , % 33L ,_1!% : < : <_1! 1 _% *olu(e of frustu( part 33, _1"! 33. 1 3₃ , % :31 4 3! 4 2313!<L

1 " 33 , % N: !  < 4 : < 4 2: " !  < 4 : ! _! < O :" M < 1! 1! 1! 1! 1! *olu(e of 1 post  ""5 % 3₃, "&33. 1 ""5 3₁ , 3₃ 4 3₃ , 1"! 4 "& *olu(e of 150 posts  !!% % 3₁ , !1" 33. !!% 3 , 1503₁ , 150 : < !1" 3 , 333. 33 33.3

9.# ooden /all ! ft. in dia(eter eighs !00 l/. Find the dia(eter of a /all of the sa(e (aterial

hih eighs 50 l/. By si(ilar

solids ₃% ₅₀

!% , _!00

3 , 3. 3333 33.

&.# ertain Chinese oin 1! in. in dia(eter and 11" in. thiK is piered /y a suare hole 1& in. on a side. Find the a(ount of (etal in a oin.

1 1 ! 1 ! 3₃, 3_�33334 3₃₃₃₃₃₃,   : < M :<& 3₃ , 0.1&09  .% 1 3 , 3₃L , 0.1&09 : < 1" ! !

=.Fro( a ylindrial glass " in. high and % in. in dia(eter ater is poured /y tilting the glass until the highest point of the /otto( of the glass lies in the plane of the ater surfae. Ho (uh ater re(ains)

6nitial $olu(e of ater+

1 _! 3₃₃₃₃ , 3₃L ,  :% <:"< 3₃₃₃₃, 1%.5  .% 1 3 , ! 3₃₃₃₃ , 1 :1%. <5 ! 3 , 3. 3333.3, 33. 333 33.3

50.Ho (any u/i feet are there in the largest piee of ti(/er of suare ross setion that an /e ut fro( the log)

a! 4 a! , %! !a! , = a! , .5 /! 4 /! , !! ! /! ,  /! , ! 3₁, 3!, .5 33.! 3_! , 3! , ! 33.!1 3 , % :31 4 3! 4 2313!<L

1

3 , !<:1&< %:.5 4 42.5

3 , 33 333

51.Ho (any /oard feet does the piee of ti(/er in :< a /oard foot /eing_{eui$alent to a /oard 1} ft. suare and 1 in. thiK)

3 , %" 33._{133. 3}33.% 3133. 333.1!11 3 , 333 33. 33.

33.

5!.Ho (any u/i feet are in the largest piee of suare ti(/er of the sa(e size throughout its

hole length)

3 , ₃3 , 3:33 , 33 33.3<

5%.# log 1& ft. long is ! ft. in dia(eter at the top end and % ft. in dia(eter at the_{/utt end.} Ho (any u/i feet of ood does the log ontain)

1 3 , % :31 4 3! 4 2313!<L , 1 1  :3!4 3!4 33<L % 3 ,1.5:1.0<O:1&< % 1.5! 4 1.0! 4 3 , 33. 3  33.3, 33. 33 33.3

5.# grain ele$ator in the for( of a frustu( of a right irular one is ! ft. high and the radii of its

/ases are 10 ft. and 5 ft. respeti$elyV ho (any /ushels of heat ill it hold if 181 u. ft. euals 1 /u.)

3 ,1 4 33< %1  :3! 4 3!

3 ,  10! 4 5! 4 10:5<O:!< %

3 , 100 33.% 3 , 100 33.%3 1 33. 1 1 33.%  3 , 3333  33. , 3333. 33 33.

55.Soap Kettles used in the o((erial (anufature of soap are as a rule large ylindrial $ats

50000 l/. or (ore of soap /eing (ade in a single /eating. Find the apaity of suh a Kettle ha$ing an inside dia(eter of 1& ft and an altitude of %0 ft. if soap eighs 90 l/. per u. ft.

*olu(e

*,#/h,1-:1&! <:%0<

*,!%0-ft.% Capaity /y eight of soap

',W*,90:!%0-<

;=534384.)1 lb

5".# fatory hi(ney is in the for( of frustu( of regular suare pyra(id. The hi(ney is 1!5 ft. high the edges of its /ases are 1! ft. and & ft.

respeti$ely. The ross8setion of the >ue is " ft. suare. Ho (any u/i feet of (aterial does the hi(ney ontain)

3 , 3_3333333 3_3333333 3 3 , 3 (34 3 4 2 3 3<3  33 3 , 1 :1!! 4 &! 4 21!! _{ &}!_{< :1!5< } "!:1!5< % 3 , 3333. 33 33.3

59.6n the foundation orK of the 'oolorth Building a 558story /uilding in Ne  UorK City it as neessary in order to reah the /edroK to penetrate the

sand and uiKsand to a depth in so(e instanes of 1%1 ft. 6f the largest irular aisson 1= ft. in dia(eter as 1%0 ft. deep and as 7lled ith onrete to ithin %0 ft of the surfae ho (any u/i yards of onrete ere reuired)

Conrete dia(eter , 1= ft. , 1=% yd.

*olu(e of onrete _{1 1=} ! 100 3 , 3₃L ,   : % < : % < 3 , 3333. 33 33.3

5&.The inside of a $ase is an in$erted one !.=&% in. aross the top and 5.01" in. deep. 6f a hea$y

sphere !.=& in. in dia(eter is dropped into it hen the $ase is full of ater ho (uh ater ill o$er>o)

tan  , 1.=15 5.01" tan₃ , 1".5"? 3 , sin  1.!= , sin 1".5"? 3 , .%&!  . 5.01"  L , 3  3 5.01"  h , .%&!  1.!= L , 1.&&%_.  

#(ount of ater that o$er>os * , seg(ent of a sphere 1 1 3 , L< , % L!:%3   :1.&&% !_{<%:1.!=<  1.&&%O} % 3 , 3. 33 33.3

Ane setion of the (ainline seer pipe for the ity of #nnapolis passes through a tunnel & ft.

long. The ross setion of the tunnel is shon in the 7gure. The seer pipe is 1" in. in e@ternal dia(eter and is partially e(/edded in a (attress of

onrete a ross setion of hih is shon in the 7gure.

5=.Find the total ost of digging the tunnel at X5.00 per u/i yard.

1 !! < :5< , 1 5 33.! 3₃ , ! :% 4 1! 1!   , & 33.

3₃ , 3₃  , 15 :& < 1! 1 33. % Cost of digging+ 3₃, 5&&.%% 33.%, 5&&.%% 33.% : 3₃, !1"."0 33.% < % 33.  ₃, 3:333. 33< , X3 333. 33 "0.Find the total ost of pouring of onrete at X=.00 per u/i yard.

3 ₉

,

5% "0 3 , "."5   .

/ , !! 4 !a , !! 4 !:".1&%< , %.%9 in. / , !.&" ft. 1 os : !  < , &  , 1!0?  1 _! 3₁, ! 3 (333 sin 333< 1 & !   3₁ , ! : <N1!0? : < M sin 1!0?O 1! 1&0? 3₁ , 0.!9% 33.! 1 9 1 9 3₃ , ! :% 4 3< : < M 3, 1 :% 4!.&"< : < M 0.!9% 1! ! 1! 3₃, 1.%9 33.! 3₃, 3₃  , 1.%9:&< , "=5."9 33.% 3₃, !5.9" 33.%

Cost of onrete pouring+

(c = ).0025.6 = <231.8)

"1.# reser$oir 10 ft. deep is in the for( of a frustu( of in$erted suare

it reuire an inlet pipe to 7ll the reser$oir if the ater pours in at the rate of !00 gal. per (in.) :Ane gal. , !%1 u. in.<

;eser$oir apaity 1 3 , % :31 4 3! 4 2313!<L 1 3 ,2100!100_:=0!_{<O:10< %}! 4 =0! 4 3 , =0%%%.%% 33.% 1!  . % 3 , =0%%%.%% 33%3 : < 1 33. 3 , 15"0="000  .% 3 1 333. !%1  .% 3 , "9590.!5=9 333.

*olu(e , isharge @ ti(e

3 M 33 "9590 , !003 3 33. 3 , 3 333. 3 333. 3 33 333. , 33. 333 333.

"!.6t is desired to ut oY a piee of lead pipe ! in. in outside dia(eter and 1 in. thiK so that it ill (elt into a u/e of edge  in. Ho long a piee ill /e reuired) ; , outer radius of pipe ; , 1 in. r, inner radius of pipe r , ; R Z , 0.95 in.

*olu(e of pipe , *olu(e of u/e

 :1!  0.95!<  , %

  , 33. 333 33.

"%.# stone is dropped into a irular tu/ 0 in. in dia(eter ausing the ater_{therein to rise !0 in.} 'hat is the $olu(e of the stone)

 The $olu(e of the stone is eual to the $olu(e of ater that rises in the tu/

1 _! 3 , 3₃L ,    :0 <:!0< 3 , &000  .% 3 , &000   .% : 1 33. < 1!  . 3 , 33. 333 33.3

".# druggist sells a ertain Kind of poder in a retangular /o@  /y ! /y 1 in. for !5 ents and in a

ylindrial an !Q in. high and !Q in. in dia(eter for 10 ents. 'hih is the (ore eono(ial /uy)

6n retangular /o@ 3 , :!.5<:1.!5< , 1!.5  .% Dnit prie 6n ylindrial an !5 3 , 1!.5 1 33333 , !  .% Dnit prie 3 ,  :!.5 !_{<:!.5< ,} 1!.!91&"%  .% 33 3 , 33. 3333333 , 3. 333 %

33333

33.3

"5.Eah of (a((oth dredges used in digging the ana(a Canal had a dipper the apaity of hih as 15 u. yd. 6f the height of the dipper as 10 ft. =Q in. hat as the area of the /otto()

3 , 15 33.%3 : % 33.<% , 05 33.% 1 33. 1 L , 10 33. = !  . , 10 4 3 , 3₃L =. 5 1 ! 1= , 10 ! 33. 1= 05 , 3₃:1010 < !  3, 33. 3333 33.3

"".#ssu(ing that a ity has 00 (iles of ater pipes and that the a$erage_{dia(eter of the pies is 1} ft. ho (uh ater is reuired to 7ll this entire syste()

3 , 3₃  3 , 1  :1!<:00 3 5!&0< , 5!&000 33.%  3 , 333 333 33.3 3 3. 33 333. , 33 333 333. 33 333. 3 33.3

"9.# onrete surge ha(/er ith irular ross setion and $ertial inner alls has an inside

dia(eter of 100 ft. The outer alls taper unifor(ly. The outer alls taper unifor(ly Z in. to 1 ft. of rise and at the /ase the thiKness is 5 ft. The height of the surge ha(/er is 150 ft. a/o$e the pressure tunnel and the (aterial used in its onstrution eighs 150 l/. per u. ft. Find the total eight of the ha(/er hen full of ater.

3

150 33.

 1 , . 

3 , %9.5  . , %.1!5 33.

3 , 55  3 , 55  %.1!5 , 51.&95 33.  Total $olu(e :ater 4 onrete<

1 ! ! 1 ! ! 3₃, 3333333 33 3 3333 , % L :3 4 3 , 1%"0%9."! 33.% 4 33< ,55 % :150< 4 51.&95 4 :55<:51.&95<O *olu(e of ater 3_ , 33�3333 , :50!<:150< , 119&0=9.!5 33. [% *olu(e of onrete 3₃ , 3₃ 3_ , 1%"0%9."!  119&0=9.!5 , 1"9=0.!19 33.% 'eight of ater '_ , W*_ , "!.:119&0=9.!5< , 9%51%!"&.0= l/ 'eight of onrete '_ , W*_ , 150:1"9=0.!19< , !551%!"&.0= l/  Total eight hen full of ater

3₃, 3₃4 3₃, 33 333. 33 33333 3333

"&.Find the aste in utting the strongest suare ti(/er fro( a irular log of unifor( dia(eter !

ft. and length & ft.

3!4 3! , !! !3! ,  3! , ! 3 _{3 3} 3 ₃ 3 3₃₃₃₃₃ , 3₃₃₃M 3₃₃₃₃₃₃ , 3 3    3   , :   3 3 <:3<  3:3< , 3. 3333 33.

"=.#n anhor ring is for(ed /y re$ol$ing a irle ! in. in dia(eter a/out a line lying in the plane of 

the irle and at a distane of 9 in. fro( its enter. Find the $olu(e of the solid for(ed. The solid for(ed :anhor ring< is generally alled torus

3 , 3₃3 2

3 , 333 � , :3  3<3 �:3< , 333. 333 33.3 90.6f a piee of /rass & /y " /y 1! in. is dran out into a ire 1%0 in. in_{dia(eter hat ill /e the}

length of the ire)

3_33 , 3₃₃₃₃₃ 1 1 :   <!  , &:"<:1!<  %0   , 333 333. 33 33. , 33 333. 33 33.

91.# oYee pot is 5 in. deep 81! in. in dia(eter at the top and 58% in. in dia(eter at the

/otto(. Ho (any ups of oYee ill it hold if " ups eual 1 uart) #nser to the nearest hole nu(/er.

1 3 , % :31 4 3! 4 2313!<L 1 3 , %  5! = "  4 &1 5!=   4 2 1" " & 1   : 1 "  < O :5< "%%5 3 , 1=!   .% , 10%."" .% "%%5 3 , 1=!   .% 3 1 333 !%1  .%  333333 3 1 333333 3 , 1.9=5 333333 Nu(/er of ups 3 , 1.9=5 333333 3 " 3333 1 33333 3 , 33. 33 , 33 3333

9!.# pipe of %8in. inside dia(eter onduts ater fro( a spring to a house %00 ft. distant. 6t is

desired to e(pty the pipe after the ater has /een turned oY the spring. 'ill a 108t. pail hold the ater) :Ane gal. , !%1 u. in.<

1 % ! 3 ,  : <  ! :%00 3 1!< 3 , 50".!5  .% 3 , 50".!5  .% 3 1 333333 !%1  .% 3 , ".&&5 3333333  333333 3 , ".&&5 3333333 3 1 333333 3 , 33. 33 333333

$here7ore/ the 10t. *ail cannot hold all the !ater in the %i%e. $he an!er i >O

9%.# spherial shell ! in. thiK has an outer dia(eter of 1! in. Find the $olu(e of the (aterial of hih it is (ade.

1 3 , :1!< , "  . ! 3 , 3  ! ,   . 3 3 3 ,  (3 33< , 3 :   33 M 33< , 333. 33 33.3 3

9.Ho (any u/i inhes of lu(/er does a stiK ontain if it is  in. /y  in. at one end ! in. /y !

in. at the other end and 1" ft. long) Dse the for(ula for

frustu( 1 3 , % :31 4 3! 4 2313!<L 3₁ ,  3  , 1"  .! 3_! , ! 3 ! ,   .! L , :1"<1! , 1=!  . 3 , 3 33 4 3 4 233:3<O :333< , 3333 33.3 3

95.# light is plaed 5 ft. fro( the enter of the glo/e % ft. in dia(eter. Find the area of the illu(inated portion. Solutio n+ os  , 1.5 _L1.5 1.5 , 5 5:1.5  L< , 1.5! 5L , 5.!5 L , 1.05 33

 3, �33 , �:3. 3<:3. 3333 33.3<

9".6n FingalIs Ca$e on the 6sland of StaYa :He/rides Sotland< the alls are for(ed /y /asalti pris(ati olu(ns $arying in height fro( 1& to %" ft.  These olu(ns (ostly he@agonal are so perfet as to suggest the hand of

(an. Ane of the $ertial olu(ns is !=.% ft. in height and has a /ase of .9 s. ft. 'hat is its $olu(e)

V=Abh V=4.7!". #$

V=137.71 ft.3

99.To spheres of lead of radii ! and % in. respeti$ely are (elted into a ylinder of re$olution of radius 1 in. Find the altitude of the ylinder. *olu(e of ylinder , *olu(e of to spheres

:1!<L ,  :!%< 4  :%%< % 3 3 , 33 3 % 33.

9&.6f the gra$ity hose ross setion is shon in the 7gure eighs 150 l/. per u. ft. 7nd the total

eight of a setion of a da( 50 ft. long. *olu(e of da(

*,#/

*,1!:41!<:15<4:<O :50< *,1%":50<

',W*

',150:"&00< ',10!0000 l/

',10!0000 l/P1 short ton!000 l/

;=510 hort ton

9=.The ater supply of a distrit onsists of a laKe hose surfae area is 1000 ares. The nu(/er of household supplied ith ater is "0000. Supposing eah household to use !5 gal. per day alulate ho (any daysI supply :to the nearest day /elo the e@at nu(/er< is represented /y a depth of 1 ft. of ater in the laKe. :1 are euals %5"0 s. ft.<

aily ater onsu(ption of "0000 households

3₃ , !5:"0000< , 1500000 333. , !005%5 33.%

*olu(e of 1 ft. epth of ater in the reser$oir 3₃, 3₃L 3₃ , 1000 3 %5"0000 33.! L , 1 33. 3₃, %5"0000:1< , %5"0000 33.% Nu(/er of days 3₃    , 3₃ 33  333 333 , 333 333 , 333 3333

&0.# fatory hi(ney has the shape of a frustu( of a regular pyra(id. 6ts height is 1&0 ft. and its

upper and loer /ases are suares hose sides are 10 ft. and 1" ft.

respeti$ely. The ross setion of the >ue is a suare hose side is 9 ft. Ho (any u/i feet of (aterial does the hi(ney ontain)

*olu(e of frustu( 1 3₃ , % :31 4 3! 4 2313!<L 1 ! ! ! ! 33 , % 1" 4 10 4 21" :1 0 <O :1&0< 3₃ , %0="0 33.% *olu(e of >ue :parallelepiped< 3₃ , 3₃L , 9! :1&0< 3₃ , &&!0 33.% *olu(e of hi(ney 3 , 3₃ 4 3₃ , %0="0  &&!0 3 , 33 333 33.3

&1.'indo glass for(erly as (ade /y gathering a lu(p of (olten glass on the end of a hollo rod and /loing it into the for( of large hollo ylinder

a/out " ft. long and 181! ft. in dia(eter. This as ut longitudinally and then plaed in an o$en and heated until it softened hen it as >attened out into plates and ut into desired sizes. 6f this plate as 0.1! in. thiK 7nd the

3 ,  �3 3 ,   :1 1 < :"< : ! 0.1! < " 3 , 3. 333333 33.3

&!.ass a plane through a u/e so that the setion for(ed ill /e a regular he@agon. 6f the edge of the u/e is ! units 7nd the area of this setion.

@!,1!41! @!,! #,":1!@!sin"0]< #,"1!:!<sin"0] O A=33? nit2

&%.Sho that the total surfae of the u/e is tie the suare of its diagonal. #,"a! #,":s% 2<! #,":s! %< A=22

&.The $olu(e of the u/e is 1=%2s% ti(es the u/e of its diagonal. *,a%

*,:s%2<%

V=1)3? 3

&5.Ho (uh (aterial as used in the (anufature of !000 elluloid die if eah die has an edge of 1 inh)

*olu(e of 1 die+

*1,:1<%_{*1,1"in%} *olu(e of !000 die

*,!000*1 *,!000:1"<

V=35 in3

&". 'hat is the eight of a /loK of ie ! in. /y ! in. /y ! in. if ie eighs =!_{per ent as (uh as ater and ater eighs "!.5 l/ per u. ft.)} ';ieice,=!^Water*24@123=0.)262.5ie

&9.Find the $olu(e and total area of the largest u/e of ood that an /e utfro( a log of irular ross setion hose radius is 1!.9 inhes. ia(eter of the log

d,!:1!.9< d,!5. in Edge of u/e a!4a!, d! !a!,!5 .! a!,%!! .5& a,19.=" in *olu(e of the largest

u/e *,a% *,19.="% *,59=%.90 in% *,59=%.90 in%P:1 ft1! in<% *,%.%5!& ft%

 Total area of the largest u/e #,"a! #,":19.="!< #,1=%5.& in!

#,1=%5.& in!P:1 ft1! in<!

A=13.441 7t2

&&.# $egeta/le /in /uilt in the for( of a u/e ith an edge of " ft. is di$ided /y a $ertial partition hih passes through to diagonally opposite edges. Find the lateral surfae of either o(part(ent.

#1,":" < #1,%"_ft! ;euired lateral area

#,!#14#! #,!:%"<4%"! 2 A=122.)1 7t2

&=.Find the area of the triangle hose $erte@ is at the (idpoint of an upperedge of a u/e of edge # and hose /ase oinides ith the diagonally_{opposite edge of the u/e.} d!,a!4_{a!d!,!a!d,a!2!}

#rea of triangle #BC

##BC,1!ad ##BC,1!a:a2 !<

AA(=1@2?a2

=0.6f a u/e has an edge eual to the diagonal of another u/e 7nd the ratio of their $olu(es.

et

a , edge of the s(aller u/e thus a2% , edge of the larger u/e

*olu(e of the s(aller u/e *s(all,a%

*olu(e of the larger u/e

*large,:a2%<_*large,%2a%%%

;euired ratio

,*s(all*large ,a%_%2a%%

=1.Ane u/e has a fae eui$alent to the total area of another u/e. Find_{the ratio of their $olu(es.} et

a , edge of s(aller u/e @ , edge of s(aller u/e

 Total area of s(aller u/e_#s(all,"a! #rea of one fae of the larger u/e #large1,@!

Ane fae of larger u/e , total area of s(aller u/e #large1,#s(all

@_@,a2!_,"a!

" *olu(e of s(aller u/e

*s(all,a% *olu(e of larger u/e

*large,@% *large,:a2" <% *large,"2a" % ;atio of $olu(es ,*s(all*large ,a%_"2a"% =?6@36

=!.Counting %& u. ft. of oal to a ton ho (any tons ill a oal /in 1= ft. long_{" ft. ide and = ft. deep ontain hen le$el full)} *,1=:"<_*,10!":=<_ft%

'eight 10!" ft% _oal

',1 ton%& ft% _:10!"ft%< ;=2 ton

=%.Co(pute the ost of lu(/er neessary to resurfae a foot/ridge 1" ft. ideand 150 ft. long ith !8in. planKs if lu(/er is X0 per 1000 /oard feet._{Neglet aste. :Ane /oard foot , 1 ft. /y 1 ft. /y 1 in.<} *,1":150<:!< ft!`(iddotinO1:1<:1<

ft!`(iddotin/oard ft.O *,&00/oard ft.  Total ost of lu(/er

C,X01000/oard ft.P&00/oard ft.

(=<1)2

=.# paKing /o@ !.! ft. /y .= ft. /y 5.5 ft. is to /e o(pletely o$ered ith tin.Ho (any suare feet of the (etal are needed) #,!:a/4/4a<

#,!!.!:.=<4.=:5.5<4!.!:5.5< O A=)).66 7t2

=5.The edges of the trunK are % ft.  ft. " ft. # seond trunK is tie as longV theother edges are % ft.  ft. Ho do their $olu(es o(pare)_*

1,%:<

:"< *ft%1,9!

*:!P"<!,%:<

*!,1 ft%

*!!*V1₂,19=2V₁

=".#n eletri refrigerator is /uilt in a for( of retangular parallelepiped. The_{inside di(ensions} are % ft. /y !." ft. /y 1.& ft. # freezing unit :1.1 ft. /y 0.& ft. /y 0.9 ft.<

su/trats fro( the storage roo( of the /o@. Find the apaity of the refrigerator.