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Mensuration Questions

41. Area of 1st circle and circumference of 2nd circle is 1386 cm2 and 176 cm respectively. There is a square whose side is 35(5/7)% of twice of sum of the radius of both the circles. Find the perimeter of the square (in cm)?




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Correct Ans:140
Explanation:
Given, Area of 1st circle = 1386 cm2
---> π r2 = 1386
---> r2 = 1386 * (7/22)
---> r2 = 63 * 7 = 441
---> r = √441
---> r = 21
Hence, radius of 1st circle = 21

Given, Circumference of 2nd circle = 176 cm
---> 2πr = 176
---> r = 176 * (7/22) * (1/2)
---> r = 4 * 7
---> r = 28
Hence, radius of 2nd circle = 28

Now, Side of the square = 35(5/7)% of 2(radius of 1st circle + radius of 2nd circle)
= (250/700) * 2(21 + 28)
= (25/70) * 2(49)
= (5/14) * 98
= 5 * 7
= 35
Side of the square = 35 cm

Then, Perimeter of the square = 4 * side = 4 * 35 = 140 cm
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42. The angles of a quadrilateral are in the ratios 2 : 4 : 7 : 5. The smallest angle of the quadrilateral equal to the smallest angle of a triangle. One of the angles of the triangle is twice the smallest angle of triangle. What is the second largest angle of triangle?




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Correct Ans:60°
Explanation:
W.K.T:- Sum of angles of quadrilateral =360°
Given, ratio of angles of quadrilateral = 2 : 4 : 7 : 5
Then, 2x + 4x + 7x + 5x = 360°
---> 18x = 360°
---> x = 20°

Hence, the Smallest angle of quadrilateral = 2x = 2 * 20° = 40°
Given, Smallest angle of a triangle = Smallest angle of quadrilateral = 40°
And, Second angle of triangle = 2 * 40° = 80°

Now, Third angle of triangle = 180° – (Sum of the other two angles of the triangle)
= 180° – (40° +80°)
= 180° – 120°
= 60°
Hence, the second largest angle of the triangle = 60°
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43. If ratio of volume of sphere to volume of cylinder is 2 : 3 and radius of cylinder is equal to the side of square whose perimeter is 84 cm then find the curved surface area of cylinder(in cm²). Given that radius of sphere is equal to radius of cylinder.




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Correct Ans:5544
Explanation:
Volume of sphere = 4/3(πr3)
Volume of the cylinder = πr2h
Volume of sphere/Volume of the cylinder = 2 : 3
[(4/3(πr3)] / (πr2h) = 2/3
2r = h
Perimeter of square = 4 * side = 84 cm
Side of square = 84/4 = 21 cm
Radius of cylinder = side of square = 21 cm
Curved surface area of cylinder = 2πrh
= 2 * (22/7) * 21 * 42 = 5544 cm2
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44. If length of a rectangle is decreased by 6 cm we get a square and the area of square formed is 252 cm2 less than the area of square formed when breadth of the original rectangle is increased by 6 cm. Find the perimeter of the rectangle.




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Correct Ans:84 cm
Explanation:
Let the length of the rectangle be 'l cm' and breadth of the rectangle be 'b cm'.
As per the question,
Area of rectangle when breadth increased - Area of reactangle when length decreased = 252
[l x (b + 6)] - [b x (l - 6)] = 252
lb + 6l - bl + 6b = 252
6(l + b) = 252
2(l + b) = 84 cm
Therefore, perimeter of rectangle is 84 cm.
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45. The barrel of a fountain pen is cylindrical in shape which radius of base as 0.7 cm and is 5 cm long. One such barrel in the pen can be used to write 300 words. A barrel full of ink which has a capacity of 14 cu.cm can be used to write how many words approximately? 




SHOW ANSWER
Correct Ans:545
Explanation:
Given:
Radius of base = 0.7 cm; Height = 5 cm

WKT, Volume of cylinder = πr2h
Volume of barrel in pen = (22/7)*(0.7)2*5
= 7.7 cu.cm
So, capacity of 7.7 cu.cm in pen can write 300 words.

Therefore, capacity of 14 cu.cm can be used to write = (300 x 14)/7.7 = 545 words.
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46. The radius of the circle is equal to seven - ninth of the side of the square.The area of the circle is 9856 sq cm. Find the perimeter of a square?




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Correct Ans:288 cm
Explanation:
Given: Radius of Circle = (7/9) * Side of Square
Area of circle = πR²
9856 = (22/7) * R²
R² = 9856 / (22/7)
= 448 * 7
R = √(448 *7)
R = 56 cm
And Side of Square = Radius of Circle * (9/7)
= 56*(9/7)
Side = 72 cm
Perimeter of the Square = 4 * Side of Square.
= 72*4
= 288 cm.
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47. The side of a rhombus is 13m. If one of its diagonal is 24m. Find the area of the rhombus. 




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Correct Ans:120 m2
Explanation:


Given: ∆BCE
BE² + EC² = BC²
BD = 24 m, BE = BD/2 = 12 m
EC = √[(13)² -(12)²] = 5 m
Diagonal AC = 2(EC) = 2*5 = 10 m
Required area = (Product of diagonals)/2
= (10*24)/2
= 120 m².
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48. A ground is square in shape has 12 m wide road inside it running along its sides. The area occupied by the road is 5290 sq m. What is the perimeter along the outer edge of the road? 




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Correct Ans:488
Explanation:
Let the side of outer square of the road = x meter.
So the side of inner square of the road = (x - 2(12)) = (x - 24) m.



According to the question:
We know that Area of square = (side)2
Area of outer square of the road - Area of inner square of the road = Area occupied by the road
x2 - (x - 24)2 = 5290 m2
x2 - x2 - 576 + 48x = 5290
48x = 5866
x = 122.2 m
Therefore side of outer square of the road = x = 122 m

Perimeter of square = 4a ,where a is area.
Perimeter of outer square of the road= 4*a = 4*122 = 488 m.
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49. The total area of a circle and a square is equal to 5450 sq.cm. The diameter of the circle is 70 cm. What is the sum of the circumference of the circle and the perimeter of the square? 




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Correct Ans:380 cm
Explanation:
Given: Diameter of circle = 70 cm
Radius of circle = 35 cm
WKT, Area of square = a2
Area of circle = πr2
Area of square + Area of circle = 5450
a2 + (22/7) x 35 x 35 = 5450
a2 + 3850 = 5450
a2 = 5450 - 3850
a2 = 1600
a = 40

As per the question,
= Circumference of the circle + Perimeter of the square
= 2πr + 4a
= 2 x (22/7) x 35 + 4(40)
= 220 + 160
= 380 cm
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50. How many tiles whose length and breadth are 12 cm and 5 cm respectively will be needed to fit in a rectangular region whose length and breadth are respectively 144 cm and 100 cm?




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Correct Ans:240
Explanation:
Total area of the rectangular region = Length * Breadth
= 144 * 100
= 14400 cm2

Area of one tile = 12 * 5 = 60 cm2

Number of tiles required = 14400 / 60 = 240
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51. A rectangular garden has a four-metre-wide road along all the four sides. The area of the road is 1104 sq metre. What is the sum of the length and the breadth of the garden?




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Correct Ans:130m
Explanation:
Formula:
Area of Rectangle = l*b
Let the length and breadth of the rectangular garden be l and b
The 4m wide road covers the inner garden.
Therefore, the length and breadth of the outer road is (l+8)*(b+8)
Given:
Area of rectangular garden including the 4m wide road = Area of the road
(l+8)*(b+8)-lb = 1104
lb+8l+8b+64-lb = 1104
8l+8b = 1104-64
l+b = 130
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52. The circumference of a circle is thrice the perimeter of a rectangle. The area of the circle is 616 sq. m. What is the area of the rectangle if the breadth of the rectangle is 60 m?




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Correct Ans:4320 sq m
Explanation:
Area of a circle= πr2
616 = 22r2 /7
616*(7/22) = r2
r2 = 196
r = 14 m
Circumference of a circle = 2πr
= 2*22/7 * 14
= 88 m2
It is given that the circumference of a circle is thrice the perimeter of a rectangle
Circumference of a circle= 3Perimeter of the rectangle
Circumference of a circle = 88 m2
= 3*88
Circumference of a circle= 264 m2
264 = 2 (l + 60)
132 = l + 60
l = 132 - 60
l = 72
Area of the rectangle = l*b
= 72*60
Area of the rectangle = 4320 m2
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53. The dimension of an open container are 100cm, 80cm, and 46cm. Its thickness is 6cm. If 1 cubic cm of metal used in the box weight 1 gram, find the weight of the container.




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Correct Ans:128.64kg
Explanation:
Volume of a Cube (V) = a3
External volume = (100*80*46) cm3
= 368000 cm3
Internal volume = (88* 68* 40) cm3
= 239360 cm3
Volume of the metal used in the container = 368000 - 239360
= 128640 cm3
Converting into g to Kg:
Weight of the metal = 128640 / 1000
= 128.64 kg
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54. The number of sperical bullets that can be made out of a solid cube of lead whose edge measures 44 cm, each bullet being of 4 cm diameter, is (take π = 22/7) 




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Correct Ans:2541
Explanation:
Given:
Bullet diameter - 4 cm; Bullet radius - 2cm
Cube edges - 44 cm

WKT, Volume of cube = a3
Volume of sphere = (4/3)πr3

Total number of spherical bullets = Volume of solid cube/Volume of 1 bullet
= (44 x 44 x 44)/[(4/3)x (22/7) x 2 x 2 x 2]
= (44 x 44 x 44 x 7 x 3)/(22 x 4 x 2 x 2 x 2)
= 2541.
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55. A horse is tied to a post by a rope. If the horse moves along a circular path always keeping the rope stretched and describes 88 m when it has traced out 72° at the center the length of the rope is (take π = 22/7)




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Correct Ans:70 m
Explanation:


WKT, Length of the arc = 2πrθ/360°
As per the question,
2πrθ/360° = 88
2πr x 72/360 = 88
r = (88 x 5)/(2 x π)
r = (88 x 5 x 7)/(2 x 22)
r = 70 m
Here, length of the rope is nothing but radius.
Therefore, length ogf the radius is 70 m.
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56. A square lawn has a path of 4m wide around it. If the area of the path is 196 m2, then each side of the lawn is




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Correct Ans:8.25 m
Explanation:


Let each side of the square lawn be 'X' m.
As per the question,
Area of lawn and path - Area of lawn = 196
(X + 4 + 4)2 - (X)2 = 196
X2 + 64 + 16X - X2 = 196
16X = 196 - 64
X = 132/16
X = 8.25 m.
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57. The diagonal of a square equals the side of an equilateral triangle. If the area of the square is 6√3 sq.cm, what is the area of the equilateral triangle?




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Correct Ans:9 sq.cm
Explanation:
Given:
Area of square = 6√3 sq.cm
a2 = 6√3 sq.cm
a = √(6√3) cm

WKT, Diagonal of square = a√2
Diagonal of square = √(12√3) cm

Area of equilateral triangle = (√3/4)a2
= (√3/4) x [√(12√3)]2
= (√3/4) x (12√3)
= 9 sq.cm
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58. A conical vessel, whose internal radius is 12 cm and height 50 cm, is full of liquid. The contents are emptied into a cylindrical vessel with internal radius 10 cm. Find the height to which the liquid rises in the cylindrical vessel.




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Correct Ans:24 cm
Explanation:
Let the height of the liquid in the conical vessel be 'h'.
Given:
Internal radius of cone =12 cm
Internal height of cone = 50 cm
Internal radius of cylinder = 10 cm

WKT, Volume of cylinder = πr2h cm3
Volume of cone = 1/3(πr2h)cm3

Volume of the liquid in the cylindrical vessel = Volume of liquid in the conical vessel
πr2h = 1/3(πr2h)
[(22/7) x 10 x 10 x h] = [(22 x 12 x 12 x 50)/(3 x 7)]
[(22/7) x 10 x 10 x h] = [(22 x 4 x 12 x 50)/7]
h = [(4 x 12 x 50)/(10 x 10)]
h = 24 cm.
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59. The area of a square is 1296 sq cm. Find the perimeter of the rectangle, if length is (1/3) the side of the square and breadth is 28 cm less than the side of the square?




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Correct Ans:40 cm
Explanation:
Given: Area of a square = 1296 sq.cm.
Area of square = a2 sq.cm
a = √1296
a = 36 cm

Length of the rectangle = (1/3) x 36 = 12 cm
Breadth of the rectangle = 36 -28 = 8 cm
Perimeter of the rectangle = 2(l + b)
= 2(12 + 8)
= 2(20)
= 40 cm.
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60. Some bricks are arranged in an area measuring 50 m3. If the length, breadth and height of each brick is 25 cm, 12.5 cm and 4 cm respectively. Find the number of bricks that are used supposing there is no gap between two bricks.




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Correct Ans:40,000
Explanation:
Given: Volume of area = 50m3
Length of brick = 25 cm = 25/100 m
Breadth of brick = 12.5 cm = 12.5/100 m
Width of brick = 4 cm = 4/100 m

WKT, Volume of each brick = l x b x h
Volume of that area = Number of bricks x Volume of each brick
50 = N x [(25 x 12.5 x 4)/(100 x 100 x 100)]
N = 50/0.00125
N = 40,000
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