Q1: The dimensions of a cuboid in cm are 16 × 14 × 20. Find its total surface area.
Ans:
Total surface area of a cuboid =2(l × b + b × h + l × h)
Let A be the total surface area of the cuboid with length =16cm, breadth =14 cm and height =20 cm
Thus,
A = 2(16 × 14 + 14 × 20 + 20 × 16)
=2(224 + 280 + 320)
=1648cm^{2}
Q2: The cuboid water tank has length 2 m, breadth 1.6m and height 1.8m. Find the capacity of the tank in litres.
Ans:
Volume of a cuboid of length l, breadth b and height h = l × b × h
So, the volume of the tank =2 × 1.6 × 1.8 = 5.76 m^{3}
One m^{3} can occupy 1000 litres.
Hence, 5.76 m^{3} can occupy 5.76 × 1000=5760 litres.
Q3: What is the length of the sheet, 2 meter wide, required for making an open tank 15 m long, 10 m wide and 5 m deep?
Ans:
Area of the sheet required to make the tank
= lb + 2bh + 2lh
=15 × 10 + 2 × 10 × 6 + 2 × 15 × 6
=450m^{2}
Length of the sheet required = Area / Width
= 450/2
m = 225m
Q4: How much length of the iron sheet 11 cm wide is required for making an open cylinder 15 cm high and 7 cm as base radius (π=22/7)?
Ans:
Surface of an open cylinder = area of the iron sheet
Let A be the surface area of open cylinder
A = πr^{2} + 2πrh
= 22/7 x 7^{2} + 2 x 22/7 x 7 + 15
= (154+660)cm^{2}
= 814cm^{2}
Sheet is of the rectangular shape whose area is given as length times breadth.
∴ Length of the sheet
= (814 ÷ 11)cm
= 74cm
Q5: Is a square prism same as a cube? Explain.
Ans: Comparing a cube with a square prism
Hence, A square prism is not a cube since its lateral faces need not be congruent to the base and top
Q6: 30 circular plates, each of radius 14 cm and thickness 3 cm are placed one above the another to form a cylindrical solid. Find the total surface area.
Ans:
Given, radius 14 cm, thickness =3 cm, number of plates = 30
Therefore, height of cylinder will be 3 × 30 = 90 cm
We know, total surface area of cylinder = 2πr(r + h) cm^{2}
Therefore, total surface area =2π(14)(14 + (90)) = 9152 cm^{2}
Q7: The total surface of a rectangular block is 846 cm^{2} . Find the volume, if the dimensions are in proportion 5 : 4 : 3.
Ans:
Let the dimensions be 5x, 4x, and 3x.
Total surface area =2(lb + bh + hl)
= 2(5x × 4x + 4x × 3x + 3x × 5x)cm^{2}
= 94x^{2} cm^{2}
Now,
94x^{2} = 846cm^{2} (given)
⇒ x^{2} = 9 or x = 3cm
∴ Volume of the block = 5x × 4x × 3x
= 60x^{3 } = 1620cm^{3}
Q8: A cone, a hemisphere and a cylinder stand on equal bases and have the same height What is the ratio of their respective volumes ?
Ans:
Let the volume of the cone, hemisphere, and cylinder be V_{1} ,V_{2} ,V_{3}
Let the radius of the cone, hemisphere, and cylinder be r
∴ height of the hemisphere is also r
= 1 : 2 : 3
Q9: Rukhsar painted the outside of the cabinet of measure 1 m × 2m × 1.5 m . How much surface area did she cover if she painted all except the bottom of the cabinet?
Ans:
A cabinet has 6 surfaces including bottom surface.
However, we have to paint only 5 surfaces.
So, we have one front and one back surface (2 × 1.5), one left and one right surface (1 × 1.5) and one upper surface (2 × 1).
So, area to be painted is
= (2 × (2 × 1.5)) + (2 × (1 × 1.5)) + (2 × 1)
= 11 sq. m
Q10: The length, breadth and height of a room are 5 m, 4 m and 3 m respectively. Find the cost white washing the walls of the room and ceiling at the rate of Rs. 7.50 per m^{2}.
Ans:
Given:
L= 5 m, B = 4 m, H = 3m
⇒ Area to be whitewashed = Area of cuboid − Area of base.
=2lh + 2bh + 2lb − lb
=2lh + 2bh + lb
= [2 × 5 × 3 + 2 × 4 × 3 + 5 × 4]m^{2}
= 74m^{2}
Cost of white washing per m^{2 } area = Rs.7.50
Cost of white washing 74m^{2} area = 74 × 7.50
= Rs.555
Q11: A road roller takes 750 complete revolutions to move once over to level a road. Find the area of the road if the diameter of a road roller is 84 cm and length is 1 m.
Ans:
Given:
⇒ D = 84 cm, l = 1m
⇒ r = 42cm
⇒ Area of road covered in 1 revolution = 2πrl = m
⇒ Area of road covered in 750 revolution =
= 1980 sq m
Q12: The floor of a rectangular hall has a perimeter 250 m. If the cost of painting the four walls at the rate of 10 per m^{2} is 15000, find the height of the hall (in meters).
Ans:
Given:
Perimeter = 250m
⇒ Area of 4 walls = 2lh + 2bh
= 2(b + l)h
⇒ Perimeter of floor of wall = 2l + 2b
⇒ 2(l + b) = 250 m
∴ Area of 4 walls =2(l + b)h
= 250h m^{2}
⇒Cost of painting per m^{2} area = Rs. 10
⇒ Cost of painting 250h m^{2} area = Rs(250h × 10)
= Rs.2500h
⇒ Painting cost of walls = Rs. 15000
⇒ 2500h = 15000
⇒ h = 15000/2500
⇒ h= 6m
Q13: A small indoor greenhouse (herbarium) is made entirely of glass panes (including base) held together with tape. It is 30 cm long, 25 cm wide and 25 cm high.
(i) What is the area of the glass?
(ii) How much of tape is needed for all the 12 edges?
Ans:
Given:
⇒ l = 30, b = 25, h = 25
(i) Total S.A of green house
=2(lb+bh+lb)
= 2(30 × 25 + 30 × 25 + 25 × 25)
= 4250 cm^{2}
(ii) For 12 edges: Total length
= 4l + 4b + 4h
= 4(30 + 25 + 25)
= 320 cm
Q14: Find the side of a cube whose surface area is 600 sq. cm
Ans:
Surface area = 600 sq.cm
⇒ 6l^{2} = 600 ...... where l is side of cube.
⇒ l = 10 cm
Side of cube = 10 cm
Q15: From a solid wooden cube of sides 14 cm a biggest hemispherical depression is carved out. What is the total surface area of the remain solid?
Ans:
Total surface area of remaining solid
= 6a^{2} −πr^{2} +2πr^{2}
= 6a^{2} +πr^{2}
= 6(14)^{2} + 22/7 x 7 x 7
= 6 x 196 + 154
= 1330 cm^{2}
Q16: A box with a lid is made of wood which is 3 cm thick. Its external length, breadth and height are 5 cm,39 cm and 30 cm respectively. Find the capacity of the box. Also find the volume of wood used to make the box.
Ans:
Volume = (56 − 6) × (39 − 6) × (30 − 6)
= 39600cm^{3}
Volume of wood =3cm×SAofexternalbox
⇒3 × 2(56 × 39 + 56 × 30 × 39 × 30)
⇒ 30204cm^{3} .
Hence, the answer is 30204cm^{3} .
Q17: Mary wants to decorate her Christmas tree. She wants to place the tree on a wooden block covered with colored paper with a picture of Santa Claus on it. She must know the exact quantity of paper to buy for this purpose. If the box has length, breadth and height as 80cm,40cm and 20cm respectively. How many square sheets of paper of side 40cm would she require?
Ans:
Quantity of paper required by Mary to cover the wooden block is equal to its total surface area which is equal to
2(80 × 40 + 40 × 20 + 20 × 80)cm^{2} = 11200 cm^{2}
Number of square sheets of paper of side 40cm = 11200 / 40 x 40 = 7
Q18: Find the volume of a sphere whose radius is 7 cm.
Ans:
Given, radius of the sphere =7 cm
We know that the volume of sphere with radius r is V = 4/3πr^{3}.
Thus,
Volume = 4/3πr^{3}
Hence, the volume of the sphere is 1437.33 cm^{3}
Q19: The three coterminous edges of a rectangular solid are 36 cm, 75 cm and 80 cm respectively, find the edge of a cube which will be of the same capacity.
Ans:
Let the edge of the cube having same capacity be a.
Volume of the cube = Volume of the cuboid
(a)^{3} = 36 × 75 × 80
= (3 × 3 × 4 × 5 × 5 × 3 × 4 × 4 × 5)
= (3 × 4 × 5)^{3}
a = (3 × 4 × 5)
= 60 cm
Q20: Two cubes each of edge 12 cm are joined end to end. Find the surface area of the resulting cuboid.
Ans:
Length of the resulting cuboid\
= (12 + 12)cm
= 24cm
Breadth and height remaining the same i.e. 12 cm each
Surface area = 2(lb + bh + lh)
= 2(24 × 12 + 12×12 + 24 × 12)^{2}
= 1440cm^{2}
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