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Mensuration Questions

1. Smallest side of a right angled triangle is 13 cm less than the side of a square of perimeter 72 cm. Second largest side of the right angled triangle is 2 cm less than the length of the rectangle of area 112 cm² and breadth 8 cm. What is the largest side of the right angled triangle?




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Correct Ans:13 cm
Explanation:
Explanation :

Side of square = 72/4 = 18 cm

Smallest side of the right angled triangle = 18 – 13 = 5 cm

Length of rectangle = 112/8 = 14 cm

Second side of the right angled triangle = 14 – 2 = 12 cm

Hypotenuse of the right angled triangle = √(25 + 144) = 13cm
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2. The length of a rectangle is 3/5th of the side of a square. The radius of a circle is equal to side of the square. The circumference of the circle is 132 cm. What is the area of the rectangle, if the breadth of the rectangle is 15 cm?




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Correct Ans:189 cm2
Explanation:
Explanation :

Circumference of the circle = 132

2Ï€R = 132;
R = 21 cm

Side of square = 21 cm

Length of the rectangle = 3/5 * 21 = 63/5

Area of the rectangle = 63/5 * 15 = 189 cm2
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3. The perimeter of a square is equal to twice the perimeter of a rectangle of length 10 cm and breadth 4 cm. What is the circumference of a semi-circle whose diameter is equal to the side of the square?




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Correct Ans:36 cm
Explanation:
Explanation :

Perimeter of square = 2(l + b)
= 2 * 2(10 + 4) = 2 * 28 = 56 cm
Side of square = 56/4 = 14 cm
Radius of semi circle = 14/2 = 7cm
Circumference of the semi-circle = 22/7 * 7 + 14 = 36 cm
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4. The length of a rectangle is reduced by 30%. By what percent would the width have to be increased to maintain the original area?




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Correct Ans:42.86%
Explanation:
Explanation :
----> Width = 30*(100/100 ) - 30
----> = (3000/70) = 42.86%
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5. Six spherical cannon balls are tightly packed into a rectangular box in one layer. Each row has two cannon balls and each column has three. What part of the box is empty?




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Correct Ans:(10/21)
Explanation:
----> Let the diameter of each ball be 2r.
----> Length of the box = 3*2r = 6r
----> Breadth = 2*2r = 4r
----> Height = 2r
----> Volume = 6r * 4r * 2r = 48r3
----> Volume of 6 balls = 6 * (4/3) * (22/7) * r3 = (176 r 3 /7)
----> The area of empty space = 48 r3 - (176 r 3/7)
----> = (160 r3/7)
----> The required fraction = ((160 r3/7)/48 r3)
----> = (10/21)
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6. A circular wire of radius 49 cm is cut and bent in the form of a rectangle whose sides are in the ratio of 4:7. The smaller side of the rectangle is ?




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Correct Ans:56cm
Explanation:
Explanation :
-----> circumference = 2*(22/7)*49 = 308 cm
-----> length of rectangle sides are 4x, 7x.
-----> circumference = 2*(4x+7x)
-----> 308 = 22x
-----> X = (308/22) = 14
-----> smaller side of rectangle = 4x = 4*14 = 56 cm
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7. In a swimming pool measuring 90 m x 40 m, 150 men take a dip. If the average displacement of water by a man is 8 m cube, What will be the rise in water level?




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Correct Ans:(1/3) m
Explanation:
Given, Average displacement of water by a man = 8 m3
Then, total volume of water displaced by 150 men = 150 * 8 m3
= 1200 m3

Total volume of the swimming pool (Volume of cuboid) = l * b * h
= 90 * 40 * h
= 3600 * h
---> (here h is the height by which the water level rises)

Now, Total volume of the swimming pool = total volume of water displaced by 150 men
---> 3600 * h = 1200
---> h = 12/36
---> h = 1/3 m
Thus, the water level rises by 1/3 m
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8. If the slant height of a right pyramid with square base is 4 metre and the total slant surface of the pyramid is 12 square metre, then what is the ratio of total slant surface and area of the base?   




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Correct Ans:(16 : 3)
Explanation:
Given, Base of the right pyramid is square
Slant height = h = 4 metre
Total slant surface area of the pyramid = 12 square metre

W.K.T: Total slant surface area (Lateral surface area) of the square Pyramid = (1/2) * Perimeter of the square base * slant height
---> 12 = (1/2) * 4a * 4
Where, a = side of the square base
---> 12 = 8a
---> a = 12/8
---> a = 3/2
Thus, the side of the square base = a = 3/2 metre
Therefore, Area of the base (i.e, area of square base) = a2
= (3/2)2
= 9/4 m2

Now, Required ratio = Total slant surface area/Area of the base
= 12/(9/4)
= (12 * 4)/9
= 16/3
= 16 : 3
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9. The base of a right prism is a quadrilateral ABCD, given that AB = 9 cm, BC = 14 cm, CD = 13 cm, DA = 12 cm and ∠DAB = 90°, If the volume of the prism be 2070 cm3, then what is the area of the lateral surface?




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Correct Ans:720 cm3
Explanation:
In ∆ABD,


Area of ∆ABD = (1/2) * AB * AD
= (1/2) * 9 * 12
= 54 cm2

In ∆ABD, by Pythagoras theorem,
BD = √[AB2 + AD2]
= √[92 + 122]
= √[81 + 144]
= √[225]
= 15 cm

In ∆BCD,
Semi-perimeter, S = (BC + CD + DB)/2
= (14 + 13 + 15)/2
= 42/2
= 21 cm

Now, Area of ∆BCD = √[S(S - a)(S - b)(S - c)]
Where, S = Semi-perimeter = 21cm
a = BC
b = CD
c = DB
Area of ∆BCD = √[21(21 - 14)(21 - 13)(21 - 15)]
= √[21(7)(8)(6)]
= √[7056]
= 84 cm2

Now, Area of quadrilateral ABCD = Area of base of Prism = Area of ∆ABD + Area of ∆BCD
= 54 + 84
= 138 cm2

Height of Prism = Volume/Area of base
= 2070/138
= 15 cm

Perimeter of base of the Prism = sum of the bases = 9 + 14 + 13 + 12 = 48 cm

Area of lateral surface = Perimeter of the base * Height of the prism
= 48 * 15
= 720 cm2
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10. 18 cylindrical water bottles with height same as that of the radius are emptied into a spherical earthen pot. The bottles fill half of the initially empty earthen pot. Find the ratio of the radius of the cylinder and that of the pot?




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Correct Ans:(1:3)
Explanation:
18 cylindrical water bottles with height same as that of the radius are emptied into a spherical earthen pot. The bottles fill half of the initially empty earthen pot. Find the ratio of the radius of the cylinder and that of the pot:

Reference:

---> Let the height and radius of bottles be r
---> Volume of all bottles = 18(πr2*r) = 18πr3
---> Let the radius of pot be P.
---> Half of the volume of the pot = (2πP3/3)
---> (2πP3/3) = 18πr3
---> πP3 = 27πr3
---> P = 3r
---> Ratio = (1:3)

Hence the answer is : (1:3)
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11. The radii of the base of a cylinder and a right circular cone are in the ratio (4 : 5) and their heights are in the ratio (2 : 3). Find the ratio of their volumes?




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Correct Ans:( 32 : 25)
Explanation:
The radii of the base of a cylinder and a right circular cone are in the ratio (4 : 5) and their heights are in the ratio (2 : 3). Find the ratio of their volumes:

Reference:

---> Volume of a cylinder = πr2h
---> Volume of a cone = (1/3)πr2h
---> Ratio of volume of cylinder to cone = (πr2h/ (1/3)πr12h1 )
---> Ratio of volume of cylinder to cone
---> = (3 * ratio of radius2 * ratio of height )
---> Given, radii of the base of a cylinder and a right circular cone are in the ratio (4 : 5) and their heights are in the ratio (2 : 3).
---> Ratio of volumes = 3 * (4/5)2 * (2/3)
---> Ratio of volumes = ( 32 : 25)

Hence the answer is : ( 32 : 25)
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12. A cone and a cylinder are of the same height. Their radii of their base are in ratio of (2 : 1). The ratio of their volumes is




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Correct Ans:(4/3)
Explanation:
A cone and a cylinder are of the same height. Their radii of their base are in ratio of (2 : 1). The ratio of their volumes is

Reference:

----> Let radius of cylinder = x and radius of cone = 2x
----> Height of each = h
----> Required ratio = (Volume of cone/Volume of cylinder)
----> = ( (1/3) π4 x2h/ π x2h )
----> = (4/3)

Hence the answer is : (4/3)
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13. The total surface area of a cylinder is 2/9 times the total surface area of a sphere. If the ratio of the curved surface area of the cylinder is 2 : 3 by the difference between the curved surface area of the cylinder and the total surface area of the top and bottom of the cylinder, find the volume of the sphere. It is given that the volume of cylinder is 3234 cubic.cm. 




SHOW ANSWER
Correct Ans:38808 cubic.cm
Explanation:
Given:
(Curved surface area of ​​the cylinder - Total surface area of ​​the top and bottom of the cylinder)/Curved surface area of ​​the cylinder = 2 : 3
WKT,Curved surface area of ​​the cylinder = 2πrh
Total surface area of top and bottom of the cylinder =2πr2
(2πrh - 2πr2)/2πrh = 2/3
2πr(r - h)/2πrh = 2/3
(h - r)/h = 2/3
1 - (r/h) = 2/3
r/h = 1/3
Let r be 'x cm' and h be '3x cm'.
Also given, volume of cylinder is 3234 cubic cm.
Wkt, Volume of cylinder =πr2h
πr2h = 3234
(22/7)*x2*3x = 3234
x3= (3234*7)/(22*3)
x3= 343
x = 7 cm
Hence, r = 7 cm; h = 21 cm

Let the radius of sphere be 'R cm'.
WKT,Total surface area of ​​a cylinder=2πrh +2πr2
Total surface area of ​​a sphere = 4πR2
Total surface area of ​​a cylinder is 2/9 times the total surface area of ​​a sphere.
2πrh +2πr2= (2/9)4πR2
2*(22/7)*7*[21 + 7] = (2/9)*4*(22/7)*R2
R2= 441
R = 21 cm.
Volume of sphere = (4/3)πR3
= (4/3) x (22/7) x 21 x 21 x 21
= 38808 cubic.cm.
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14. A room is of dimensions 9*8*6.5 in meters. It has one door of dimensions 2 m x 1.5 m and three windows each of dimensions 1.5 m x 1 m. Find the cost of white washing the walls but excluding the door and windows, at Rs 3.80 per square metre.




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Correct Ans:Rs. 811.3
Explanation:
Given, Length of the room = 9 m
Breadth of the room = 8 m
Height of the room = 6.5 m
W.K.T: Lateral surface area of a Cuboid = 2 (l + b) * h
Area of the 4 walls of room = 2 (l + b) * h
= 2 (9 + 8) * 6.5
= 34 * 6.5
= 221 m2

Area of 1 door = 2 * 1.5 = 3 m2

Area of 1 window = 1.5 × 1 = 1.5 m2
Area of 3 windows = 3 × 1.5 = 4.5 m2

Area required to white wash the wall = Area of 4 walls - Area of 1door - Area of 3 windows
= 221 - 3 - 4.5
= 213.5 m2

Cost of white washing the walls = 213.5 * 3.8
= Rs. 811.3
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15. The sum of circumference of a circle and perimeter of a rectangle is 480 cm. If the area of the rectangle is 168 cm2 and ratio of radius of the circle and breadth of the rectangle is 19 : 2. Find the area of another circle whose radius is 28% of the diagonal of rectangle. (Note: Length and Breadth of rectangle must be integer)




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Correct Ans:154 cm2
Explanation:
Given:
Area of the rectangle = 168 cm2
Ratio of radius of the circle and breadth of the rectangle = 19 : 2
Let the radius of the circle be 19x and breadth of the rectangle be 2x.
WKT, Area of rectangle = l*b
l*b = 168
l*2x = 168
l = 84/x cm

WKT, circumference of a circle = 2π r
Perimeter of rectangle = 2(l + b)
Sum of circumference of a circle and perimeter of a rectangle = 480 cm
2π r + 2(l + b) = 480
2*(22/7)*19x + 2[(84/x) + 2x] = 480
36x2 - 140x + 49 = 0
x = 7/2
So, breadth of rectangle = 2(7/2) = 7 cm
length of rectangle = 84/(7/2) = 24 cm
Diagonal of rectangle = √[242 + 72]
= √[576 + 49]
= √625
= 25 cm
Let the radius of new circle be 'R'.
Radius of new circle is 28% of the diagonal of rectangle = (28/100)*25
= 7 cm.
Area of new circle = π R2
= (22/7)*(7)2
= 154 cm2.
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16. If the sum of the length, breadth and height of a rectangular parallelepiped is 24 cm and the length of its diagonal is 15 cm, then its total surface area is




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Correct Ans:351 cm2
Explanation:
If the sum of the length, breadth and height of a rectangular parallelepiped is 24 cm and the length of its diagonal is 15 cm, then its total surface area is

Reference :

----> l + b +h = 24
----> l2 + b2 + h2 = 15 2
----> l2 + b2 + h2 = 225
----> (l+b+h)2 = l2 + b2 + h2 + 2(lb+bh+hl)
----> 242 = 225 + 2(lb+bh+hl)
----> 2(lb+bh+hl)=576 – 225 = 351 cm2

Hence the answer is : 351 cm2
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17. Two steel sheets each of length a1 and breadth a2 are used to prepare the surfaces of two right circular cylinders "“ one having volume v1 and height a2 and other having volume v2 and height a1. Then,




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Correct Ans:a2v1 = a1v2
Explanation:
Area of sheets = a1a 2
Area of sheets = Area of cylinder
-----> a1a 2 = 2πrh
-----> For first cylinder, height = a2
-----> Therefore,
-----> a1a 2 = 2πr1a2
-----> r1 = (a1/2π)
-----> Similarly, for the second cylinder,
-----> r2 = (a2/2π)
-----> The ratio of volumes , ((π r12a2)/(π r22a1)) = (v1/v2)
-----> (v1/v2) = (a1/a2)
-----> (v1/a1) = (v2/a2)

Hence the answer is : a2v1 = a1v2
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18. Two cones have their heights in the ratio 1:3 and the radii of their bases in the ratio 3:1. Find the ratio of their volumes?




SHOW ANSWER
Correct Ans:3:1
Explanation:
Ratio of volume = (1/3πr12h1) / (1/3πr22h2)
We know that,
Height = 1 : 3
Radius = 3 : 1
Ratio of volume = 9/1 * 1/3 = 3/1 = 3 : 1
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19. A sector of 120 degrees, cut out from a circle, has an area of 66/7 sq cm. Find the radius of the circle?




SHOW ANSWER
Correct Ans:3 cm
Explanation:
πr2 * 120°/360° = 66/7
22/7 * r2 * 1/3 = 66/7
r2 = 3*3
r = 3 cm
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20. A spherical lead ball of radius 6 cm is melted and small lead balls of radius 3 mm are made. The total number of possible small lead balls is




SHOW ANSWER
Correct Ans:8000
Explanation:
Formula:
Volume of sphere = (4/3)Ï€r3
1 cm = 10 mm
A spherical lead ball of radius 6 cm is melted and small lead balls of radius 3 mm are made.
Radius of small lead balls = 3/10 = 0.3 cm
Let n be the number of small lead balls
Therefore,
(4/3)Ï€(6)3 = n * (4/3)Ï€(0.3)3
(6*6*6*1000)/(3*3*3) = n
n = 8000
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