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Mensuration Questions

21. The cross section of a canal is in the shape of an isosceles trapezium which is 4 m wide at the bottom and 5 m wide at the top. If the depth of the canal is 2 m and it is 120 m long, what is the maximum capacity of this canal?




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Correct Ans:1080 cubic mts
Explanation:
The canal is 4 m wide at the bottom and 5 m at the top, thus the canal is in the shape of a trapezium whose volume will be equal to the product of area of trapezium and the width.
Length of canal = 120 m
Depth of canal = 2 m
Crosse sectional area of canal = 1/2*120*(4 + 5) = 540 sq.m.
Maximum capacity of canal = cross sectional area * height
= 540 * 2
= 1080 cubic mts
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22. The perimeter of a rectangular plot is 48 m and area is 108 .The dimensions of the plot are




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Correct Ans:18 m and 6 m
Explanation:
The perimeter of a rectangular plot is 48 m and area is 108 .The dimensions of the plot are

Reference:

Let find the length and breath:

------->P = 2(l+b)
------->2(l+b) = 48
------->l+b = 24----(i)
------->A = l * b
------->l * b = 108----(ii)
------->By solving (i) and (ii), we get:
------->l = 18 m and b = 6 m.
------->18 m and 6 m

Hence the answer is : 18 m and 6 m
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23. If the radius of a cylinder is doubled and the height remains same, the volume will be




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Correct Ans:Four times
Explanation:
Let the volume of cylinder be V = πr2h

If radius is doubled, new radius = 2r

New volume = π(2r)2h = 4πr2h

New Volume = 4 × Volume

Hence, the correct answer is :Four times
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24. A conical cavity is drilled in a circular cylinder of 15 cm height and 16 cm base diameter. The height and the base diameter of the cone are the same as those of the cylinder. Determine the total surface area of the remaining solid. 




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Correct Ans:440π cm2
Explanation:
Total surface area = CSA of cylinder + Area of base of cylinder + CSA of cone
= 2πrh + πr2 + πrl
= 240π + 64π + 136π
= 440π cm2
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25. A hemisphere is made of a sheet of a metal 1 cm thick. If the outer radius is 5 cm, what is the weight of the hemisphere (1 cm3 of the metal weigh 9 g)?




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Correct Ans:366π g
Explanation:
Thickness the sheet of metal = 1 cm
Volume of sheet = 2/3π(5)3 - 2/3π(4)3
= 122π/3
Weight of the hemisphere = (122π*9)/3
= 366π
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26. A trapezium based prism with two parallel sides 8 cm and 14 cm respectively and distance between two parallel sides is 8 cm. Find the height of the prism if the volume of the prism is 1056 cm³ ? 




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Correct Ans:12 cm
Explanation:
Given:
Volume of the prism = 1056 cm³
WKT, Volume of prism = Base x Height
Base area = Area of trapezium
Area of trapezium = (1/2)(a + b)*h = (1/2)[8 + 14]*8
= (1/2)*22* 8 = 88 cm2

Volume of prism = (Area of trapezium) x Height
1056 = 88 * height
Height of prism = 12 cm.
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27. If the area of a circle, inscribed in an equilateral triangle is 4π cm², then what is the area of the triangle? 




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Correct Ans:12√3 cm2
Explanation:
Given:
Area of a circle = 4π cm²
πr2 = 4π
r2 = 4
r = 2 cm.
WKT, Radius of incircle = a/2√3
2 = a/2√3
a = 4√3.

WKT, Area of equilateral triangle = (√3/4)a2
Area of equilateral triangle = (√3/4)(4√3)2
= 12√3 cm2.
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28. A rectangular water tank is 80 m x 40 m. Water flows into it through a pipe of 40 sq.cm at the opening at a speed of 10 km/hr. The water level will rise in the tank in half an hour is




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Correct Ans:(5/8) cm
Explanation:
Reference:
A rectangular water tank is 80 m x 40 m. Water flows into it through a pipe of 40 sq.cm at the opening at a speed of 10 km/hr. The water level will rise in the tank in half an hour is

Volume of water filled by pipe in 30 minutes

-----> (40 *1000000/2) cu.cm
-----> = 2000000 cu.cm

Therefore, Height of water level

-----> (2000000/8000 *4000)
-----> = (5/8) cm.

Hence the answer is: (5/8) cm
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29. Two circular cylinders of equal volume have their heights in the ratio 1: 2. Ratio of their radii is




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Correct Ans:(√2:1)
Explanation:
The ratio of volumes = ((πr12h1)/(πr22h2)) = 1

According to question, h1: h2 = (1: 2)

Therefore,
-------> ((r12)/(r22) )* (1/2)
-------> ((r1)/(r2)) = (√2/1)


Hence, the correct answer is : (√2 : 1)
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30. The length, breadth and height of a room is 5m, 4m and 3m, respectively. Find the length of the largest bamboo that can be kept inside the room




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Correct Ans:5√2 m
Explanation:
Anything related to the largest rod, bamboo or anything to be fitted inside a cube or cuboid is its body diagonal.
Body diagonal of a cube = a√3
Body diagonal of a cuboid = √l2 + b2 + h2
Length of largest bamboo
= √52 + 42 + 32
= √25 + 16 + 9 = √50 = 5√2 m
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31. Each side of an equilateral triangle is 6 cm. Find its area.




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Correct Ans:9√3 sq.cm.
Explanation:
Area of triangle = √S (S - a) (S - b) (S - c)
Where s = (a + b + c)/2
Here a = b = c = 6
Thus S = 18/2 = 9
Area of triangle = √9 (9 - 6) (9 - 6) (9 - 6)
= √9 * 3 * 3 * 3 = 9√3 sq.cm.
Hence option A is correct
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32. The theatre in the town is built in the form of a kite. It's perimeter is 100 m. If one of the sides measure 20 m, then what are the measurements of others sides?




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Correct Ans:20 m, 30 m, 30 m
Explanation:
W.K.T, the shape of Kite resembles Quadrilateral.
The adjacent sides of a kite are of equal length.

Given, perimeter of Quadrilateral = 100 m
Length of one of the side = x = 20 m
Let the Length of other side = y

W.K.T, perimeter of Quadrilateral = sum of the length of all the four sides
---> 100 = 20 + 20 + y + y
---> 100 = 40 + 2y
---> 60 = 2y
---> y = 30 m

Hence, the length of sides of the kite (ie., Quadrilateral) = 20 m, 20 m, 30 m, 30 m
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33. The area of a field in the shape of a trapezium measures 1440 m2. The perpendicular distance between the parallel side is 24m. If the ratio of the parallel sides is 5:3, what is the length of the longer parallel side?




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Correct Ans:75 m
Explanation:
Given, ratio of the parallel sides of a trapezium = 5 : 3
---> Let the parallel sides (ie., 'a' and 'b') be 5x and 3x respectively.
Given, perpendicular distance = height = 24 m
and Area = 1440 m2

Formula: Area of Trapezium = (1/2) * h * (a + b)
---> 1440 = (1/2) * 24 * (5x + 3x)
---> 1440 = 12 * (8x)
---> 120 = 8x
---> 15 = x

Thus, parallel sides, a = 5x = 5 * 15 = 75 m
and b = 3x = 3 * 15 = 45 m
Hence, Longer parallel side is 'a' = 75 m
Smaller parallel side is 'b' = 45 m.
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34. A circle is inscribed in a square. If the difference between area of the square and circle is 262.5 cm², then find the area of the rectangle whose perimeter is same as that of circle while length of rectangle is 20% more than the breadth of rectangle (in cm²). 




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Correct Ans:750
Explanation:
Let the side of square be 2a.
So, the diameter of circle = 2a
Radius of circle = a

WKT, Area of square = 4a
Area of circle = π r2
Perimeter of circle = 2π r
Area of rectangle = l x b
Perimeter of rectangle = 2(l + b)
Given that difference between area of the square and circle is 262.5 cm².
Area of square - Area of circle = 262.5
4a2 - (22/7)a2 = 262.5
28a2 - 22a2 = 1837.5
6a2 = 1837.5
a2 = 306.25
a = 17.5 cm

Perimeter of circle = 2πa = 2 x (22/7) x 17.5 = 110 cm

Also given length of rectangle is 20% more than the breadth of rectangle.
Let the breadth of rectangle be X.
So, length of rectangle = (120/100)X = 1.2X
As per the question,
Perimeter of rectangle = Perimeter of circle
2(1.2X + X) = 110
2.4X + 2X = 110
4.4X = 110
X = 25 cm
Breadth of rectangle = 25 cm
Length of rectangle = 1.2(25) = 30 cm

Area of rectangle = 25 x 30 = 750 cm2.
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35. There were two solid spherical balls. Ratio between radius of first ball to that of second ball is 4 : 3. Second ball was cut into two equal halves and the difference between total surface area of first ball and total surface area of a part of second ball is 1424.5 cm². Find value of radius of bigger ball ? 




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Correct Ans:14 cm
Explanation:
Given:
Ratio between radius of first ball to that of second ball is 4 : 3.
Let the radius of first ball be '4r' and radius of second ball be '3r'.
When we cut the second ball it become hemisphere.

WKT, Total surface area of sphere = 4π r2
Total surface area of hemisphere = 3π r2


As per the question,
4 x (22/7) x (4r)2 - 3 x (22/7) x (3r)2 = 1424.5
(22/7)*r2[64 - 27] = 1424.5
r2 = [1424.5*7]/[37*22]
r2 = 12.25
r = 3.5 cm
Therefore, radius of bigger ball = 4r = 4(3.5) = 14 cm.
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36. Radius of a cylinder is equal to the side of an equilateral triangle having area 16√3 cm² and height of the cylinder is equal to the perimeter of the triangle. Then find the volume of cylinder.




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Correct Ans:1536π cu.cm
Explanation:
Given:
Area of equilateral triangle = 16√3 cm²
WKT, Area of equilateral triangle = √3/4 a2
√3/4 a2 = 16√3
a2 = 64
a = 8 cm

Side of an equilateral triangle = Radius of a cylinder
a = r
r = 8cm
Perimeter of equilateral triangle = 3a = 3(8) = 24 cm

Perimeter of equilateral triangle = Height of cylinder
So, h = 24 cm
Volume of cylinder = πr2h
= π x 8 x 8 x 24
= 1536π cm3.
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37. The volume of a brick is 3000 cm3 and its dimensions are in the ratio 3 : 2 : 4. If its entire surface is painted at 50 paisa per cm2. The cost will be (in Rs.) 




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Correct Ans:Rs. 650
Explanation:
Let the length of the brick be 3X, breadth of the brick be 2X and height of the brick be 4X.
Given: Volume of a brick is 3000 cm3
WKT, Volume of brick = l x b x h
Volume of brick = 3X*2X*4X
3000 = 24X3
X = 5
Length of the brick = 3X = 3(5) = 15 cm
Breadth of the brick = 2X = 2(5) = 10 cm
Height of the brick = 4X = 4(5) = 20 cm

WKT, Total surface area of brick = 2(lb + bh + hl)
Total surface area of brick = 2[(15 x 10) + (10 x 20) + (20 x 15)]
= 2[150 + 200 + 300]
= 1300 2
The cost = 1300 x (50/100) = Rs. 650.
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38. If the radius of sphere is increased by 9.09 %, then surface area of sphere is increased by how much percentage. 




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Correct Ans:19%
Explanation:
Let the radius of sphere be r1.
Increased radius of sphere, r2 = r1 + (9.09/100)r1
= 12r1/11
WKT, Surface area of sphere = 4π r2
Surface area = 4π r12
Increased surface area = 4π r22 - 4π r12
= 4π[(12r1/11)2 - r12]
= 92πr12/121
Percentage increase in surface area = [(92πr12/121)/4π r12]*100
= (23/121)*100
= 19%.
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39. A special type of cylindrical vessel with radius and height of 24.5 cm and 5 cm respectively is used to hold Cognac. The vessel is filled upto 80% of its capacity and then total Cognac from cylindrical vessel transferred to 9 cuboidal vessels whose length and breadth is 7 cm & 8 cm respectively. Find the height of each cuboidal vessel? 




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Correct Ans:14.9 cm
Explanation:
Given:
Cylinder: Radius - 24.5 cm; Height - 5 cm
Cuboid: Length - 7 cm; Breadth - 8 cm
WKT, Volume of cylinder = πr2h
Volume of cubiod = l x b x h

Volume of 80% of Cognac in cylindrical vessel = Volume of 9 cuboidal vessels
(22/7) x 24.5 x 24.5 x 5 x 0.8 = 9 x 7 x 8 x h
h = [(22/7) x 24.5 x 24.5 x 5 x 0.8 ]/[9 x 7 x 8]
h = 7546/504
h = 14.9 cm.
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40. The length and breadth of a rectangle are in the ratio of 9:5. When the sides of the rectangle are extended on each side by 5 m the ratio of length to breadth becomes 5:3. What is the area of the original rectangle? 




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Correct Ans:1125 sq.m
Explanation:
Let the length and breath of rectangle be 'X' m and 'Y' m respectively.
Given, length and breadth of a rectangle are in the ratio of 9 : 5.
X/Y = 9/5
5X = 9Y
5X - 9Y = 0 ....(i)

If sides of the rectangle are extended by 5 m, ratio becomes 5 : 3.
(X + 5)/(Y + 5) = 5/3
3(X + 5) = 5(Y + 5)
3X + 15 = 5Y + 25
3X - 5Y = 10 ....(ii)
By solving equation (i) and (ii),
X = 45 m ; Y = 25 m
So, length of rectangle = 45 m
Breadth of rectangle = 25 m

WKT, Area of rectangle = l x b
Area of rectangle = 45 x 25 = 1125 sq.m.
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