Mensuration Additional Questions-10th Std Maths-Book Back Question And Answer
I. Multiple Choice Questions:
Question 1.
The curved surface area of a right circular cylinder of radius 1 cm and height 1 cm is equal to ______
(1) π cm2
(2) 2π cm2
(3) 3π cm2
(4) 2 cm2
Answer:
(2) 2π cm2
Hint:
C.S.A of a cylinder = 2πrh sq. units = 2 × π × 1 × 1 cm2 = 2π cm2
Question 2.
The total surface area of a solid right circular cylinder whose radius is half of its height h is equal to ______ sq. units.
(1) 32πh
(2) 23πh2
(3) 32πh2
(4) 23πh
Answer:
(3) 32πh2
Hint:
T.S.A = 2πr(h + r)
[radius is half of the height]
= 2π×h2(h2+h)
= =πh(3h2)=32πh2 sq. units
Question 3.
Base area of a right circular cylinder is 80 cm2. If its height is 5 cm, then the volume is equal to _______
(1) 400 cm3
(2) 16 cm3
(3) 200 cm3
(4) 4003 cm3
Answer:
(1) 400 cm3
Hint:
Volume of a cylinder = πr2h cu. units
[Base area (πr2) = 80 cm2 = 80 × 5 cm3 = 400 cm3
Question 4.
If the total surface area of a solid right circular cylinder is 200π cm2 and its radius is 5 cm, then the sum of its height and radius is ______
(1) 20 cm
(2) 25 cm
(3) 30 cm
(4) 15 cm
Answer::
(1) 20 cm
Hint:
T.S.A of a cylinder = 200π cm2
2πr (h + r) = 200π
2 × 5 (h + r) = 200
(h + r) = 20 cm
Question 5.
The curved surface area of a right circular cylinder whose radius is a units and height is b units, is equal to ______
(1) πa2b sq.cm
(2) 2πab sq.cm
(3) 2π sq.cm
(4) 2 sq.cm
Answer:
(2) 2πab sq.cm .
Hint:
C.S.A. of a cylinder = 2πrh sq. units = 2 × π × a × b sq. cm = 2πab sq. cm
Question 6.
Radius and height of a right circular cone and that of a right circular cylinder are respectively, equal. If the volume of the cylinder is 120 cm3, then the volume of the cone is equal to _______
(1) 1200 cm3
(2) 360 cm3
(3) 40 cm3
(4) 90 cm3
Answer:
(3) 40 cm3
Hint:
Volume of the cone = 13 × volume of the cylinder
= 13 × 120 cm3
= 40 cm3
Question 7.
If the diameter and height of a right circular cone are 12 cm and 8 cm respectively, then the slant height is
(1) 10 cm
(2) 20 cm
(3) 30 cm
(4) 96 cm
Answer:
(1) 10 cm
Hint:
Slant height of a cone
Question 8.
If the circumference at the base of a right circular cone and the slant height are 120π cm and 10 cm respectively, then the curved surface area of the cone is equal to ______
(1) 1200π cm2
(2) 600π cm2
(3) 300π cm2
(4) 600 cm2
Answer:
(2) 600π cm2
Hint:
Circumference (2πr) = 120π cm
Slant height (l) = 10 cm;
Curved surface area of a cone = πrl sq. units
= 120π2 × 10 cm2 = 600π cm2
Question 9.
If the volume and the base area of a right circular cone are 48π cm and 12π cm respectively, then the height of the cone is equal to ______
(1) 6 cm
(2) 8 cm
(3) 10 cm
(4) 12 cm
Answer:
(4) 12 cm
Hint:
Volume of a cone = 48π cm3
[Base area (πr2) = 12π]
13 πr2h = 48π
13 × 12π × h = 48π
[Substitute πr2 = 12π]
h = 484 = 12 cm
Question 10.
If the height and the base area of a right circular cone are 5 cm and 48 sq.cm respectively, then the volume of the cone is equal to _______
(1) 240 cm3
(2) 120 cm3
(3) 80 cm3
(4) 480 cm3
Answer:
(3) 80 cm3
Hint:
Volume of a cone (V) = 13 πr2h sq. units
Base area (πr2) = 48 sq. cm
V = 13 × 48 × 5 = 80 cm3
Question 11.
The ratios of the respective heights and the respective radii of two cylinders are 1 : 2 and 2 : 1 respectively. Then their respective volumes are in the ratio _______
(1) 4 : 1
(2) 1 : 4
(3) 2 : 1
(4) 1 : 2
Answer:
(3) 2 : 1
Hint:
h1 : h2 = 1 : 2
r1 : r2 = 2 : 1
Ratio of their volumes
= 13πr21h1:13πr22h2
= 22 × 1 : 12 × 2 = 4 : 2 = 2 : 1
Question 12.
If the radius of a sphere is 2 cm, then the curved surface area of the sphere is equal to ________
(1) 8π cm2
(2) 16 cm2
(3) 12π cm2
(4) 16π cm2
Answer:
(4) 16π cm2
Hint:
C.S.A of a sphere = 4πr2 sq. units
[radius = 2 cm]
= 4 × π × 22 cm2
= 16π cm2
Question 13.
The total surface area of a solid hemisphere of diameter 2 cm is equal to _______
(1) 12 cm2
(2) 12π cm2
(3) 4π cm2
(4) 3π cm2
Answer:
(4) 3π cm2
Hint:
Radius of a hemisphere = 22 = 1 cm
Total surface area of a hemisphere = 3πr2 sq. units = 3 × π × 12 cm2 = 3π cm2
Question 14.
If the volume of a sphere is 916π cu.cm, then its radius is ________
(1) 43 cm
(2) 34 cm
(3) 32 cm
(4) 23 cm
Answer:
(2) 34 cm
Hint:
Volume of the sphere = 916π
Question 15.
The surface areas of two spheres are in the ratio of 9 : 25. Then their volumes are in the ratio _______
(1) 81 : 625
(2) 729 : 15625
(3) 27 : 75
(4) 27 : 125
Answer:
(4) 27 : 125
Hint:
Ratio of their surface area = 9 : 2
Question 16.
The total surface area of a solid hemisphere whose radius is a units, is equal to ________
(1) 2πa2 sq. units
(2) 3πa2 sq. units
(3) 3πa sq. units
(4) 3a2 sq. units
Answer:
(2) 3πa2 sq. units
Hint:
T.S.A. of a solid hemisphere = 3πr2 sq. units
= 3 × π × a × a sq.units
= 3πa2 sq. units
Question 17.
If the surface area of a sphere is 100π cm2, then its radius is equal to ______
(1) 25 cm
(2) 100 cm
(3) 5 cm
(4) 10 cm
Answer:
(3) 5 cm
Hint:
Surface area of a sphere = 100π cm2
4πr2 = 100π
r2 = 25
r = √25 = 5 cm
Question 18.
If the surface area of a sphere is 36π cm2, then the volume of the sphere is equal to _______
(1) 12π cm3
(2) 36π cm3
(3) 72π cm3
(4) 108π cm3
Answer:
(2) 36π cm3
Hint:
Surface area of a sphere = 36π cm2
4πr2 = 36π
r2 = 9
r = 3 cm
Volume of a sphere = 43πr3 cu. units
= 43π × 3 × 3 × 3 cm3 = 36π cm3
Question 19.
If the total surface area of a solid hemisphere is 12π cm2 then its curved surface area is equal to ______
(1) 6π cm2
(2) 24π cm2
(3) 36π cm2
(4) 8π cm2
Answer:
(4) 8π cm2
Hint:
T.S.A of a hemisphere = 12π cm2
3πr2 = 12π
r2 = 4
r = 2
Curved surface area of a hemisphere = 2πr2 = 2 × π × 4 = 8π cm2
Question 20.
If the radius of a sphere is half of the radius of another sphere, then their respective volumes are in the ratio _____
(1) 1 : 8
(2) 2 : 1
(3) 1 : 2
(4) 8 : 1
Answer:
(1) 1 : 8
Hint:
r1=r22⇒r1r2=12⇒r1:r2=1:2
II. Answer the following questions:
Question 1.
Curved surface area and circumference at the base of a solid right circular cylinder are 4400 sq.cm and 110 cm respectively. Find its height and diameter.
Answer:
Given, Circumference of the base of a cylinder = 110 cm
2πr = 110 ……. (1)
Curved surface area = 4400 cm2
2πrh = 4400 cm2 ……. (2)
From (1) & (2), (2)(1)⇒2πrh2πr=4400110=40cm
Height of the cylinder (h) = 40 cm
From (1), 2πr = 110
2 × 227 × r = 110
r = 352
We know that, diameter (d) = 2 × radius
d = 2 × 352 = 35 cm
Diameter of the Circular cylinder = 35 cm
Question 2.
A mansion has 12 right cylindrical pillars each having radius 50 cm and height 3.5 m. Find the cost of painting the lateral surface of the pillars at ₹ 20 per square metre.
Answer:
Given, Radius of a cylinder (r) = 50 cm = 0.5 m
Height of a cylinder (h) = 3.5 m
Curved surface area of a pillar = 2πrh sq. units
Curved surface area of 12 pillars = 12 × 2πrh
= 12 × 2 × 227 × 0.5 × 3.5 m2
= 132 sq. m.
Cost for painting the lateral surface of pillars per metre = ₹ 20
Cost of painting = 132 × ₹ 20 = ₹ 2640
Question 3.
The total surface area of a solid right circular cylinder is 231 cm2. Its curved surface area is two thirds of the total surface area. Find the radius and height of the cylinder.
Answer:
Given, Total surface area of a cylinder (T.S. A) = 231 sq.cm
Curved surface area = 23 × T.S.A = 23 × 231 = 154 cm2
2πrh = 154 cm2 …… (1)
Total surface area = 231 cm2
2πr (h + r) = 231
2πrh + 2πr2= 231
154 + 2πr2 = 231 [from (1)]
2πr2 = 231 – 154 = 77
Radius of the cylinder = 3.5 cm
Height of the cylinder = 7 cm
Question 4.
The total surface area of a solid right circular cylinder is 1540 cm2. If the height is four times the radius of the base, then find the height of the cylinder.
Answer:
Given, Let the radius of the cylinder be ‘r’
Height of a cylinder (h) = 4r (by given condition)
Total surface area = 1540 cm2
2πr(h + r) = 1540 c
Height of the cylinder = 4r = 4 × 7 = 28 cm
Question 5.
If the vertical angle and the radius of a right circular cone are 60° and 15 cm respectively, then find its height and slant height.
Answer:
Given, In the figure, OAB is a cone and OC ⊥ AB
∠AOC = 60∘2 = 30°
In the right ∆OAC, tan 30° = ACOC
Slant Height of the cone (l) = 15 × 2 = 30 cm
Question 6.
The central angle and radius of a sector of a circular disc are 180° and 21 cm respectively. If the edges of the sector are joined together to make a hollow cone, then find the radius of the cone.
Answer:
Given, Radius of a sector (r) = 21 cm
The angle of the sector (θ) = 180°
Let “R” be the radius of the cone.
Circumference of the base of a cone = Arc length of the sector
Radius of the cone (R) = 10.5 cm
Question 7.
If the curved surface area of a solid hemisphere is 2772 sq.cm, then find its total surface area.
Answer:
Given, the Curved surface area of a solid hemisphere = 2772 cm2
2πr2 = 2772
Total surface area = 3πr2 sq. units
= 3 × 227 × 21 × 21
= 4158 sq.cm
Aliter:
C.S.A of a hemisphere = 2772 cm2
2πr2 = 2772 cm2
πr2 = 27722 = 1386 cm
T.S.A of a hemisphere = 3πr2 sq.units = 3 × 1386 cm2 = 4158 cm2
Question 8.
An inner curved surface area of a hemispherical dome of a building needs to be painted. If the circumference of the base is 17.6 m, find the cost of painting it at the rate of ₹ 5 per sq. m.
Answer:
Given, Circumference of the dome = 17.6 m
2πr = 17.6
r=17.6×72×22=8.8×722=2.8m
Curved surface area of the dome = 2πr2 sq. units
= 2 × 227 × 2.8 × 2.8 m2
= 49.28 m2
Cost of painting for one sq.metre = ₹ 5
Cost of painting the curved surface = 49.28 × ₹ 5 = ₹ 246.40
Question 9.
Volume of a solid cylinder is 62.37 cu.cm. Find the radius if its height is 4.5 cm.
Answer:
Given, Height of a cylinder (h) = 4.5 cm
Volume of a solid cylinder = 62.37 cu. cm
Radius of a cylinder (r) = 2.1 cm
Question 10.
A rectangular sheet of metal foil with dimension 66 cm × 12 cm is rolled to form a cylinder of height 12 cm. Find the volume of the cylinder.
Answer:
After rolling the rectangular sheet into a cylinder
Volume of the cylinder = 4158 cm3
Question 11.
The circumference of the base of a 12 m high wooden solid cone is 44 m. Find the volume.
Answer:
Given, Height of the wooden solid cone (h) = 12 m
Circumference of the base = 44 m
2πr = 44
r = 44×72×22 = 7 m
Volume of the wooden solid = 13πr2h cu. units
= 13×227×7×7×12m3
= 88 × 7
= 616 m3
Volume of the solid = 616 m3
Question 12.
Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 14 cm.
Answer:
Given, Edge of the cube = 14 cm
The largest circular cone is cut out from the cube.
Radius of the cone (r) = 142 = 7 cm
Height of the cone (h) = 14 cm
Volume of a cone
Volume of a cone = 718.67 cm3
Question 13.
The thickness of a hemispherical bowl is 0.25 cm. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl. (Take π = 227)
Answer:
Let r, R and w be the inner and outer radii and thickness of the hemispherical bowl respectively.
Given that r = 5 cm, w = 0.25 cm
R = r + w = 5 + 0.25 = 5.25 cm
Now, outer surface area of the bowl = 2πR2
= 2 × 227 × 5.25 × 5.25
= 173.25 sq. cm
Thus, the outer surface area of the bowl = 173.25 sq. cm
Question 14.
Volume of a hollow sphere is 113527 cm3. If the outer radius is 8 cm, find the inner radius of the sphere. (Take π = 227)
Answer:
Let R and r be the outer and inner radii of the hollow sphere respectively.
Let V be the volume of the hollow sphere.
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Hence, the inner radius r = 5 cm
Question 15.
How many litres of water will a hemispherical tank whose diameter is 4.2 m?
Answer:
Radius of the tank = 4.22 = 2.1 m
Volume of the hemisphere
= 23πr3 cu.units
= 23×227×2.1×2.1×2.1m3
= 19.404 m3
= 19.404 x 1000 lit
= 19,404 litres
III. Answer the following questions.
Question 1.
A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
Answer:
For cylindrical part:
Radius (r) = 7 cm
Height (h) = 6 cm
Curved surface area = 2πrh = 2 × 227 × 7 × 6 cm2 = 264 cm2
For hemispherical part:
Radius (r) = 7 cm
Surface area (h) = 2πr2
= 2 × 227 × 7 × 7 cm2
= 308 cm2
Total surface area = (264 + 308) = 572 cm2
Question 2.
A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.
Answer:
Question 3.
From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm2.
Answer:
For cylinderical part:
Height (h) = 2.4 cm
Diameter (d) = 1.4 cm
Radius (r) = 0.7 cm
Total surface area of the cylindrical part
For conical part:
Base area (r) = 0.7 cm
Height (h) = 2.4 cm
Question 4.
A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m. Find the height of the platform.
Answer:
Diameter of the cylindrical well = 7 m
Radius of the cylinder (r) = 72 m
Depth of the well (h) = 20 m
Volume = πr2h
= 227×72×72×20m3
= 22 × 7 × 5 m3
Volume of the earth taken out = 22 × 7 × 5 m3
Now this earth is spread out to form a cuboidal platform having
Length (l) = 22 m
Breadth (b) = 14 m
Let ‘h’ be the height of the platform.
Volume of the platform = 22 × 14 × h m3
Volume of the platform = Volume of the earth taken out
22 × 14 × h = 22 × 7 × 5
h=22×7×522×14=52m=2.5m
Thus, the required height of the platform is 2.5 m.
Question 5.
The perimeters of the ends of the frustum of a cone are 207.24 cm and 169.56 cm. If the height of the frustum is 8 cm, find the whole surface area of the frustum. [Use π = 3.14]
Answer:
Let the radii of circular ends are R and r [R > r]
Perimeter of circular ends are 207.24 cm and 169.56 cm
2πR = 207.24 cm
The whole surface area of the frustum = π [(R2 + r2) + (R + r) l]
Required whole surface area of the frustum
= 3.14 [332 + 272 + (33 + 27) × 10] cm2
= 3.14 [1089 + 729 + 600] cm2
= 3.14 [2418] cm2
= 7592.52 cm2
Question 6.
A cuboid-shaped slab of iron whose dimensions are 55 cm × 40 cm × 15 cm is melted and recast into a pipe. The outer diameter and thickness of the pipe are 8 cm and 1 cm respectively. Find the length of the pipe. (Take π = 227)
Answer:
Let h1 be the length of the pipe
Let R and r be the outer and inner radii of the pipe respectively.
Iron slab:
Volume = lbh = 55 × 40 × 15 cm3
Iron pipe:
Outer diameter, 2R = 8 cm
Outer radius, R = 4 cm
Thickness, w = 1 cm
Inner radius, r = R – w = 4 – 1 = 3 cm
Now, the volume of the iron pipe = Volume of the iron slab
Time is taken by the pipe to empty half of the tank = 3 hours 12 minutes.
Question 7.
The perimeter of the ends of a frustum of a cone are 44 cm and 8.4π cm. If the depth is 14 cm., then find its volume.
Answer:
Given let the radius of the top of the frustum be “R” and the radius of the bottom of the frustum be “r”
Question 8.
A tent is in the shape of a right circular cylinder surmounted by a cone. The total height and the diameter of the base are 13.5 m and 28 m. If the height of the cylindrical portion is 3 m, find the total surface area of the tent.
Answer:
Given, Total height of solid = 13.5 cm
Diameter of the cylinder (d) = 28 m
Height of a cylinder (h) = 3 m
Height of a conical portion = 13.5 – 3 = 10.5 m
From the diagram, Radius of a cone = Radius of a cylinder
Radius (r) = 14 m
