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Mensuration Questions

61. A well of 11.2 m diameter is dug 8 m deep. The earth taken out has been spread all round it to a width of 7 m to form a circular embankment. Find the height of this embankment.




SHOW ANSWER
Correct Ans:1.97
Explanation:
Given:
Well diameter = 11.2 m; Radius = 5.6 m
Width of circular embankment = 7m

Volume of earth taken out = πr2h
= (22/7) x 5.6 x 5.6 x 8
= 788.48 m 3

Area of circular embankment = Area of (well + embankment) - Area of well
= π(5.6 + 7)2 - π(5.6)2
= [(22/7) x (12.6)2] - [(22/7) x (5.6)2]
= 498.96 - 98.56
= 400.4 m2.

Height of embankment = Volume of earth taken out/Area of embankment
= 788.48/400.4
= 1.97 m.
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62. If the ratio of curved surface area of cylinder to the volume of cylinder is 2 : 21 while the ratio of diameter of cylinder to the height of cylinder is 7:3. Find the total surface area of cylinder?




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Correct Ans:5148 m2
Explanation:
Formula:
Total surface area of cylinder = 2*π*r2 + 2*π*rh
Given:
Curved surface area of cylinder:Volume of cylinder = 2:21
Diameter of cylinder: Height of cylinder = 7:3
To find the height:
i.e, d/h = 7/3
It is known that Diameter (d) = 2* radius (R)
2R/h = 7/3
R/h = 7/6
h=6R/7
It is given that:
Curved surface area of cylinder/Volume of cylinder = 2/21
2πRh / πR2h = 2/21
2πR(6R/7) / πR2(6R/7) = 2/21
R = 21
h= (6*21)/7
h=18
Total surface area of cylinder = 2 * (22/7) * 21(21+18)
= 132 * 39
Total surface area of cylinder = 5148 m2
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63. The water in a rectangular reservoir having a base 80 m by 60 m is 6.5 m deep. In what time can the water be emptied by a pipe of which the cross-section is a square of side 20 cm, if the water runs through the pipe at the rate of 15 km per hour?




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Correct Ans:52 hrs
Explanation:
Volume of water in the reservoir = (80*60* 13/2)
= 31299 m3
Volume Of water flown in 1 hour = (15 * 1000 * 20/100 * 20/100) = 600 m3
Time taken = (31200/600) = 52 hrs
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64. If the breadth of a parallelogram is increased by 30% while the height of the parallelogram is decreased by 20% then find percentage change in area of the parallelogram?




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Correct Ans:4% increased
Explanation:
Let the breadth and height of the parallelogram is 10 cm and 10cm,
Normal area = 10*10 = 100
New length = 10*130/100 = 13
New height = 10*80/100 = 8
New area = 13*8 = 104
Required percentage = [(104 - 100)/100]*100 = 4 % increased
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65. The difference between the areas of a rectangle and square is 35 cm2. If the rectangle's length and breadth are 50% more and 10% less respectively than the side of the square, what is the area of the rectangle ? (in cm2)




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Correct Ans:135
Explanation:
Side of square = x cm.
Length of rectangle = 150x/100 = 3x/2 cm.
Its breadth = 90x/100 = 9x/10 cm.
According to the question,
3x/2 * 9x/10 - x2 = 35
27x2/20 - x2 = 35
27x2 - 20x2 = 35*20
7x2 = 35*20
x2 = 35*20/7 = 100
x = 10 cm
Area of rectangle = 3x/2 * 9x/10 = 27x2/20
= 27*100 / 20 = 135 sq. cm.
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66. If the breadth of a parallelogram is increased by 30% while the height of the parallelogram is decreased by 20% then find percentage change in area of the parallelogram?




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Correct Ans:4% increased
Explanation:
Area of Parallelogram = base* height
Assume that,
breadth of the parallelogram = 10cm
height of the parallelogram = 10 cm
Normal area= 10*10 = 100
New length= 10*130/100 = 13
New height= 10*80/100 = 8
New area= 13*8 = 104
Required percentage = [(104 - 100)/100]*100 = 4% increased
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67. The area of square is 196 sq. cm. whose side is half the radius of a circle. The circumference of the circle is equal to breadth of a rectangle. If perimeter of the rectangle is 712 cm, what is the length of the rectangle ?




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Correct Ans:180 cm
Explanation:
Side of square,
a2 = 196
a = √196 = 14 cm
Radius of circle = (14*2) = 28 cm
Circumference of circle = 2πr
= 2*(22/7)*28 = 176 cm
Perimeter of the rectangle = 2(l + b)
If the length of rectangle be x cm then,
2(x + 176) = 712
x + 176 = 712/2 = 356
x = 356 - 176 = 180 cm.
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68. The ratio of diameter and height of a right circular cylinder is 4 : 3. If diameter of the cylinder get reduced by 25% then its total surface area reduced to 2079 sq. cm. What is the circumference of the base of the cylinder ?(in cm)




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Correct Ans:28 π
Explanation:
Radius of cylinder = 2x cm,
New radius = 2*75% = 1.5x cm.
Height of cylinder = 3x cm.
According to the question,
Total surface area of cylinder
= 2πrh + 2πr2
= 2πr (h + r)
=2π*15x(3x+1.5x)
= 3πx * 4.5x
= 13.5πx2 sq. cm.
13.5πx2 = 2079
x2 = 2079/(13.5π)
= (2079*7) / (13.5*22) = 7*7
x = 7
Circumference of the base of cylinder = 2πr = 2π * 14
= 28π cm.
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69. The area of a square is 225 cm√2, which is equal to the area of a rectangle. The length of a rectangle is 16 cm more than the breadth of the rectangle. What is the respective ratio between the side of square and breadth of rectangle? 




SHOW ANSWER
Correct Ans:5:3
Explanation:
Given that:
Area of square = 225
side√2 = 225
Side = 15 cm A
Area of rectangle = Area of square

Let the breadth = b
Area = b(b+16) = 225
b√2 + 16b - 225=0
b√2+25b-9b- 225=0
b(b+25) - 9(b+25)=0
(b-9) (b + 25) = O
b=9
Hence ratio = side/breadth = 15/9
= 5/3
Hence ratio = 5:3
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70. The area of a rectangle is equal to the area of a square whose diagonal is 12√6 metre. The difference between the length and the breadth of the rectangle is 6 metre. What is the perimeter of rectangle ? (in metre).




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Correct Ans:84 metre
Explanation:
Area of square = (diagonal)2/2
= (12√6 * 12√6)/2
= 432 m2
Area of rectangle = 432 m2
Breadth of rectangle = x metre
length = (x + 6) metre
(x + 6)*x = 432
x2 + 6x - 432 = 0
x2 + 24x - 18x - 432 = 0
(x - 18)(x + 24) = 0
x = 18, -24
x = 18 metre
Perimeter of rectangle = 2(x + 6 + x)
= 4x + 12
= 4*18 + 12 = 72 + 12
Therefore, perimeter of rectangle = 84 metre
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71. If the areas of the three adjacent faces of a cuboidal box are 120 cm square, 72 cm square and 60 cm square respectively, then the volume of the box is:




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Correct Ans:720 cm cube
Explanation:
Area of 3 adjacent faces = lb*bh*lh
= (lbh)2
(lb*bh*lh) = (120*72*60)
(lbh)2 = (120*72*60)
lbh = √12*10*12*6*6*10
= (12*10*6)
Volume = 720cm3
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72. A tank is 7 m long and 4 m wide. At what speed should water run through a pipe 5 cm broad and 4 cm deep so that in 6 hours and 18 minutes, the water level in the tank rises by 4.5 m?




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Correct Ans:10 kmph
Explanation:
Formula:
Volume = Length * Width * Height
Here,
L = 7 m
W = 4 m
H =
Volume of water flown = (7*4* 9/2)
= 126 m2
Let the speed of the water be x km/hr = (x*1000* 63/10)* 5/100 * 4/100
= 126
x = (5*126)/63
=10
Speed of water = 10km/hr
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73. A goat is bound at one corner of a square field of side 20 m. If the length of rope is 7/10 time of that of side of field then find area grazed by goat.




SHOW ANSWER
Correct Ans:154 m2
Explanation:
Given , side of the square = 20 m
Length of rope = (7/10) * side of the square field
= (7/10) * 20
= 14 m



Then, Area grazed by goat = (1/4)th of the area of circle
= (1/4) * π * r2
here, r = Length of rope = 14 m
Area grazed by goat = (1/4) * (22/7) * 142
= 154 m2
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74. The radius of a circle is three fifth of that of a sphere. The volume of the sphere is 4000Ï€/3 m³. If ratio between area of a square and that of the circle is 14: 11, then what is the perimeter of the square (Use π = 22/7).




SHOW ANSWER
Correct Ans:48 m
Explanation:
W.K.T: volume of Sphere = (4/3) π r3
where.r = radius of sphere.
Given that, volume of sphere = 4000π/3 m3
----> (4/3) π r3 = 4000π/3
----> r3 = 1000
----> r = 10 m

Given that, radius of a circle = (3/5) * radius of sphere
= (3/5) * 10
= 6 m

Given, Area of a square : Area of the circle = 14: 11
---> a2 : πr2 = 14 : 11
---> a2/ (π * 62) = 14/11
---> a2/ (36π) = 14/11
---> a2 = (14/11) * 36π
---> a2 = (14/11) * 36 * (22/7)
---> a2 = 144
---> a = √144
---> a = 12 m ---> which is the side of the square.

Perimeter of the square = 4 * a = 4 * 12 = 48 m.
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75. The base of a solid right prism is a triangle whose sides are 9 cm, 12 cm, and 15 cm. The height of the prism is 5 cm. Then, the total surface area of the prism is __________. 




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Correct Ans:288 cm square
Explanation:
Perimeter of triangle = a+b+c
= 9 + 12+ 15
= 36 cm
Area of triangle = bh/2
=(1/2) x 9 x 12
= 54 sq. cm
Total surface area of the prism = Perimeter of base x height + 2 x Area of base
= 36 * 5 + 2 * 54
= 288 sq. cm
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76. A well with 21 meter diameter is dug 10m deep. Earth taken out of it, has been evenly spread all around it to a width of 14 m to form an embankment. The height (in meters) of the embankment is :




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Correct Ans:2(1/4)
Explanation:


Given, Diameter of the well = 21 m
And Height of the well, H = 10 m
Width of embankment = 14 m

let height of embankment = h meter
Since the well is cylindrical in shape,
Then, Volume of earth taken out (embankment) = Volume of the well
---> π*(R2 - r2) * h = πr2H
---> {[(14 + 21 + 14)/2]2 - [21/2]2} * h = (21/2)2 * 10
---> {[(49)/2]2 - [21/2]2} * h = (21/2)2 * 10
----> (1/2)2 {492 - 212} * h = (1/2)2 (212) * 10
---> {2401 - 441} * h = 4410
---> 1960 * h = 4410
---> h = 441/196
Dividing both numerator and denominator by 49, we get
---> h = 9/4
---> h = 2(1/4) meter
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77. A rectangular lawn 80 m × 60 m has two roads each 10 m wide running in the middle of it, one parallel to the length and the other parallel to the breadth. Find the cost of gravelling them at Rs. 30 per square metre.




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Correct Ans:Rs. 39000
Explanation:
Given, Rectangular lawn length = 80 m
Breadth = 60 m



Area to be gravelled = Area(ABCD) + Area(PQRS) - Area(EFGH)
= (80 * 10) + (60 * 10) - (10 * 10) m2
= 800 + 600 - 100 m2
= 1300 m2

Cost of gravelling the two roads = Area to be gravelled * Cost per square metre
= 1300 * 30
= Rs. 39,000
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78. The length of a rectangle is twice its breadth. If its length is decreased by 5 cm and breadth is increased by 5 cm, the area of the rectangle is increased by 75 sq. cm. find the length of the rectangle. 




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Correct Ans:40 cm
Explanation:
Let breadth of the rectangle be 'Y'
So, the length of the rectangle = 2Y
WKT, Area of rectangle = l x b
Area of rectangle = 2Y x Y

If its length is decreased by 5 cm and breadth is increased by 5 cm,
Then, new area of rectangle = (2Y - 5) x (Y + 5)

As per the question,
(2Y - 5) (Y + 5) - (2Y) (Y) = 75
2Y² + 10Y - 5Y - 25 - 2Y² = 75
5Y = 75 + 25
5Y = 100
Y = 20 cm

Therefore, length of the rectangle = 2Y = 2(20) = 40 cm.
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79. The area of a sector of a circle of radius 36 cm is 72π cm. The length of the corresponding arc of the sector is 




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Correct Ans:4π cm
Explanation:
Given:
Radius = 36 cm
Area of sector of a circle = 72π cm

WKT, Area of sector = (π r² θ)/360°
(π r² θ)/360° = 72π
θ = (72π x 360°) /(π x r²)
θ = (72 x 360°)/(36 x 36)
θ = 20°

Length of arc = (2π r θ)/360°
= (π r θ)/180°
= (π x 36 x 20θ)/180°
= 4π cm.
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80. The length of a rectangle is 4/7 of the side of a square. The radius of a circle is equal to the side of the square. The circumference of the circle is 396 cm and the breadth of the rectangle is 6cm.What is area of the rectangle?




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Correct Ans:216 cm2
Explanation:

Let the side of the square = 'a'
Given, length of a rectangle = (4/7) * a
Radius of a circle (r) = a
circumference of the circle = 396 cm
----> Circumference of the circle = 2πr= 396
----> 2 * (22/7) * r = 396
----> r = (396 * 7) / (2 * 22)
----> r = 63 cm

Given, Radius of a circle (r) = side of the square = 63 cm
length of a rectangle= (4/7) * side of the square
= (4/7) * 63
= 36 cm

Given Breadth of the rectangle = 6 cm
Then, Area of the rectangle = Lenghth * Breadth
= 36 * 6
= 216 cm2

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