10th Grade > Mathematics > Area
SURFACE AREAS AND VOLUMES MCQs
:
A
Volume of iron
=(440×260×100) cm3.
Internal radius of the pipe
=30 cm.
External radius of the pipe
=(30+5) cm=35 cm.
Let the length of the pipe be h cm.
Volume of iron in the pipe = (External volume) - (Internal volume)
=[π(35)2×h−π(30)2×h]cm3
=(πh)((35)2−(30)2) cm3
=(65×5)πh cm3
Volume of iron in the pipe = Volume of iron block
⇒(325πh) cm3=440×260×100
h=440×260×100325×722 cm
h=11200 cm
h=11200100=112 m
Hence, the length of the pipe is 112 m.
:
C
Height of Cylinder =h=AO=2.4 cm
Diameter of Cylinder =1.4 cm
Radius of Cone = Radius of Cylinder
OB=r=1.42=0.7 cm
Lets find Slant Height l of cone, using pythagoras theorem on △AOB , we get
AB=l=√h2+r2=√(2.4)2+(0.7)2
l=√5.76+0.49=√6.25=2.5 cm
Surface Area of remaining Solid
= Surface Area of the Cylinder + Inner surface Area of the hollow Cone
=(2πrh+πr2)+πrl=πr(2h+r+l)=227×0.7(2×2.4+0.7+2.5)=2.2(4.8+0.7+2.5)=2.2(8)=17.60 cm2
∴ Total surface area of the remaining solid is 17.60 cm2
Anita buys a new salt cellar in the shape of a cylinder topped by a hemisphere as shown below. The cylinder has a diameter of 6 cm and a height of 10 cm. She pours the salt into the salt cellar, so that it takes up half the total volume of the cellar. Find the depth of the salt, marked with x in the diagram
:
C
Let the depth of the salt in the cellar be ′x′
Volume of = volume of cylinder +
salt cellar volume of hemisphere
=πr2h+23πr3=π×32×10+23π×33=π[90+18]=108π cm3
So height x will come on the cylinder.
Half of total volume =54π cm3
πr2x=54ππ×32×x=54π9x=54⇒x=6
∴ The salt will be 6 cm deep.
:
B
Surface area = base circle + curved surface area of the cylinder + curved surface area of the cone.
=πr2+2πrh+πrl
=π(5)2+2π(5)(20)+π(5)(11)
=π×5(5+40+11)
=227×5×56
=880 cm2.
:
A
Radius of the conical vessel, R=AC=6cm
Height of the conical vessel, h=OC=8cm
Radius of the sphere, PD=PC=r
∴PC=PD=rAC=AD=6 cm
[Since, lengths of two tangents from an external point to a circle are equal]
△OCA & △OPD are right triangle.
[∵ Tangent and radius are perpendicular to each other]
OA=√OC2+AC2=√82+62 =√100=10 cmOP2=OD2+PD2
OD=OA−AD=10−6=4 cmOP=OC−PC=8−r (8−r)2=42+r264−16r+r2=16+r216r=48⇒r=3 cm.
Volume of water overflown = Volume
of sphere
=43πr3=43π×(3)3=36π cm3
Original volume of water = volume of
cone
=13πr2h=13π×62×8=96π cm3
∴ Fraction of water overflown =Volume of water overflownOriginal volume of water=36π96π=38=0.375
∴ Fraction of water overflown is 0.375
:
B
Let R be the radius of the given cone, r the radius of the small cone, h be the height of the frustum and h1 be the height of the small cone.
In the figure,△ONC∼△OMA
∴ONOM=NCMA [Sides of similar triangles are proportional]
⇒h140=rR
⇒h1=(rR)×40 .......(i)
We are given that Volume of small conevolume of given cone=164
⇒13πr2×h113πR2×40=164
⇒r2R2×140×[(rR)40]=164 [ By (i) ]
⇒(rR)3=164=(14)3
⇒ rR=14 ...... (ii)
From (i) and (ii) h1=14×40=10 cm
∴,h=40−h1=(40−10) cm
⇒h=30 cm
Hence, the section is made at a height of 30 cm above the base of the cone .
:
D
S.A. = 1372 cm2 (Given)
⇒2(l×b+b×h+l×h)=1372
[∵ Surface area of a cuboid of dimensions l×b×h is given by 2(lb+bh+lh).]
Also, it is given that l:b:h=4:2:1.
Let l=4k, b=2k and h=k.
Then, 2(4k×2k+2k×k+4k×k)=1372
⟹28k2=1372
⟹k2=49
⟹k=7
∴l=4k=4×7=28 cm
:
A
CSA of a cylinder =2πrh
Given,
CSA =88 cm2 and h=14 cm
2πrh=882r=88πh =88×722×114 =4×7×1142r=2 cm
Therefore, diameter = 2r = 2 cm.
:
D
Surface area of sphere =
4πr2=616⇒r2=616×722×14⇒r2=49⇒r=√49⇒r=7 cm
∴ The diameter, D=2r=2×7=14 cm
:
Radius of first Sphere = r1=6cm
Radius of second Sphere = r2=8cm
Radius of third Sphere = r3=10cm
Let radius of resulting sphere = r cm
According to given condition, we have
Volume of 1st sphere + Volume of 2nd sphere + Volume of 3rd sphere = Volume of resulting sphere
⇒43.π.(r1)3+43.π.(r2)3+43.π.(r3)3=43.π.r3
⇒43((r1)3+(r2)3+(r3)3)=43.r3
⇒ (63+83+(10)3)=r3
⇒ ( 216 + 512 + 1000 ) = r3
⇒ r3 = 1728
⇒ r = 12 cm