BANK & INSURANCE (MENSURATION) PART 3

Total Questions: 60

51. Perimeter of a rectangle is x meter and circumference of a circle is 52 meter more than the perimeter of the rectangle. Ratio of radius of circle and length of the rectangle is 3: 4 and ratio of length and breadth of rectangle is 7:3. Find the area of the rectangle?

Correct Answer: (c) 336 Sq m
Solution:

Perimeter of rectangle = 2 × (l + b) = x
2l + 2b = x and 2πr = x + 52

r/l = 3/4 and l/b = 7/3
r : l : b = 21 : 28 : 12 (21y, 28y, 12y)

2πr = x + 52
2πr = 2l + 2b + 52

πr = l + b + 26

(22/7) × 21y = 28y + 12y + 26
66y = 40y + 26

26y = 26
y = 1

Length of the rectangle = 28y = 28 m
Breadth of the rectangle = 12y = 12 m

The area of the rectangle = l × b = 28 × 12 = 336 sq m

52. If the volume of the cone is 1232 cm³ and the area of the rectangle is 360 cm² and the perimeter of the square is 60 cm. If the side of the square is equal to the breadth of the rectangle and the length of the rectangle is equal to the height of the cone, then what is the slanting height of the cone?

Correct Answer: (e) None of these
Solution:

Side of the square = 60/4 = 15 cm
Breadth of the rectangle = 15 cm
Length of the rectangle = 360/15 = 24 cm

Height of the cone = 24 cm
1232 = 1/3 × 22/7 × r × r × 24

Radius of the cone = 7 cm
Slanting height of the cone = √(24² + 7²) = 25 cm

53. The radius and height of a cylinder are 15 cm and 20 cm respectively. If the radius is increased by __ % and the height is increased by __ %, the volume of the cylinder would increase by __ %. Which of the following values can we fill in the same order?

(i) 25, 20, 87.5
(ii) 30, 10, 85
(iii) 10, 40, 70
(iv) 20, 25, 80

Correct Answer: (d) Only (i) and (iv)
Solution:

Suppose the radius of the cylinder is R and height is H.
Volume = πR²H

(i) New volume = π(1.25 R)²(1.2 H) = 1.875 πR²H
Increase in volume = 87.5%

(ii) New volume = π(1.3 R)²(1.1 H) = 1.859 πR²H
Increase in volume = 85.9%

(iii) New volume = π(1.1 R)²(1.4 H) = 1.694 πR²H
Increase in volume = 69.4%

(iv) New volume = π(1.2 R)²(1.25 H) = 1.8 πR²H
Increase in volume = 80%

Only (i) and (iv) satisfy the condition

54. Area of the circle is 154 cm² and the radius of the cone is equal to the radius of the circle. Height of the cone is double of the height of the cylinder whose volume is 7392 cm³ and the curved surface area of the cone is 550 cm².

From the statement given in the above question which of the following can be determined:

A. Slanting height of the cone
B. Radius of the cylinder
C. Total surface area of the cylinder
D. If the radius of the cylinder is equal to the breadth of the rectangle whose perimeter is 68 cm, then find the area of the rectangle?

Correct Answer: (d) All A, B, C and D
Solution:

Area of the circle = 154 cm
22/7 × r × r = 154
Radius of the circle = 7 cm

550 = 22/7 × l × 7
Slanting height of the cone = 25 cm
Height of the cone = √(25² − 7²) = 24 cm

Height of the cylinder = 24/2 = 12 cm

7392 = 22/7 × R × R × 12
Radius of the cylinder = 14 cm

Total surface area of the cylinder = 2 × 22/7 × 14 × (12 + 14)
= 2288 cm²

2 × (14 + l) = 68 ⇒ l = 20
Area of the rectangle = 20 × 14 = 280 cm²

55. Radius of a right circular cylinder is equal to length of a rectangle having area ___ cm² and length of the rectangle is 2 cm more than its breadth. If the height of the cylinder is equal to side of a square having area ___ cm² and the volume of the cylinder is 44616 cm³. Which of the following satisfies the two blanks given in the questions?

I. 624 cm², 441 cm²
II. 728 cm², 484 cm²
III. 143 cm², 7056 cm²

Correct Answer: (a) Both I and III
Solution:

From I: Let the breadth of the rectangle = b cm
Length = (b + 2) cm

Area of rectangle = length × breadth
⇒ 624 = (b + 2) × b
⇒ b² + 2b − 624 = 0
⇒ (b − 24)(b + 26) = 0

⇒ b = 24, −26 (rejected)
⇒ Breadth = 24 cm
⇒ Length = 24 + 2 = 26 cm = Radius of the cylinder

Let the side of the square = n cm
Area of square = (side)²
⇒ 441 = n²
⇒ n = √441
⇒ n = 21 cm

Side of the square = Height of the cylinder = 21 cm

Volume of cylinder = πr²h
⇒ (22/7) × 26 × 26 × 21
⇒ 44616 cm³

This is satisfies the given condition.

From II: Let the breadth of the rectangle = b cm
Length = (b + 2) cm

Area of rectangle = length × breadth
⇒ 728 = (b + 2) × b
⇒ b² + 2b − 728 = 0
⇒ (b − 26)(b + 28) = 0

⇒ b = 26, −28 (rejected)
⇒ Breadth = 26 cm
⇒ Length = 26 + 2 = 28 cm = Radius of the cylinder

Let the side of the square = n cm
Area of square = (side)²
⇒ 484 = n²
⇒ n = √484
⇒ n = 22 cm

Side of the square = height of the cylinder = 22 cm

Volume of cylinder = πr²h
⇒ (22/7) × 28 × 28 × 22
⇒ 54208 cm³

This is not satisfies the given condition.

From III: Let the breadth of the rectangle = b cm
Length = (b + 2) cm Area of rectangle = length × breadth
=> 143 = (b + 2) × b
=> b² + 2b - 143 = 0
=> (b - 11) (b + 13) = 0
=> b = 11, -13 (rejected)
=> Breadth = 11 cm
=> Length = 11 + 2 = 13 cm = radius of the cylinder

Let the side of the square = n cm
Area of square = (side)²
=> 7056 = n²
=> n = √7056
=> n = 84 cm

Side of the square = height of the cylinder = 84 cm
Volume of cylinder = πr²h
=> 22/7 × 13 × 13 × 84
=> 44616 cm³

This is satisfying the given condition.

56. The height of the cylinder is equal to the perimeter of the square whose diagonal is ___ m and the radius of the cylinder is equal to the side of the square and the volume of the cylinder is ___.

Correct Answer: (c) 7√2, 4312
Solution:

From Option (a) Diagonal
= a√2, 14√2 = a√2 => a = 14

Perimeter of square = 4 × 14 = 56
Volume of the cylinder = (22/7) × 14 × 14 × 56
= 34496

This not satisfies the given condition.

From Option (b) Diagonal = a√2 = 21
√2 = a√2 => a = 21

Perimeter of square = 4 × 21 = 84
Volume of the cylinder = (22/7) × 21 × 21 × 84
= 116424

This not satisfies the given condition.

From Option (c) Diagonal = a√2
a√2 = 7√2 => a = 7

Perimeter of the square = 4 × 7 = 28
Volume of the cylinder = (22/7) × 7 × 7 × 28
= 4312

This satisfies the given condition.

57. If the ratio of the radius of the circle to the cylinder is 1: 2 and the height of the cylinder is 10 cm more than that of the radius of the cylinder. The radius of the cone and height is radius of the circle and height of the cylinder respectively and the circumference of the circle is 44 cm. From the statement given in the above question which of the following can be determined.

A. Volume of the cylinder
B. Length of the cone
C. Curved surface area of the cone
D. Total surface area of the Cylinder

Correct Answer: (a) All A, B, C and D
Solution:

Circumference of the circle is 44 cm.
44 = 2 × (22/7) × r
Radius of the circle = 7 cm

The ratio of the radius of the circle to the cylinder is 1 : 2
So, Radius of the cylinder = (2/1) × 7 = 14 cm

Height of the cylinder = 14 + 10 = 24 cm
Volume of the cylinder = πr²h
(22/7) × 14 × 14 × 24 = 14784 cm³

TSA of the cylinder = 2πr (h + r)
2 × (22/7) × 14 (14 + 24) = 3344 cm²

Radius of the cone = Radius of the circle = 7 cm
Height of the cone = Height of the cylinder = 24 cm

Length of the cone = √(24² + 7²) = 25 cm
CSA of the cone = πrl = 22/7 × 7 × 25 = 550 cm²

We can find answers of all the given questions.

58. If the ratio of the height of the cone to the height of the cylinder is 3: 1 and the ratio of the radius of the cone to cylinder is 1: 1. The area of the rectangle is 280 cm² and the length of the rectangle is 6 cm more than that of the breadth of the rectangle. Height of the cone is 20% of the diagonal of the rectangle. From the statement given in the above question which of the following can be determined.

A. Volume of the Cone
B. Perimeter of the rectangle
C. Curved surface area of the cylinder
D. Total surface area of Cone

Correct Answer: (e) Only B
Solution:

The area of the rectangle = lb
280 = (b + 6) × b
b² + 6b = 280
b² + 6b - 280 = 0
b² + 20b - 14b - 280 = 0
b(b + 20) - 14(b + 20) = 0
(b - 14)(b + 20) = 0
b = 14, -20 (negative value neglect)

Length of the rectangle = 14 + 6 = 20
Perimeter of the rectangle = 2(20 + 14) = 68 cm

Diagonal of the rectangle = √596 cm
Height of the cone = √596 × 20/100 = √596/5 cm

Height of the cylinder = (√596/5) × (1/3) = √596/15 cm

We cannot find the radius of the cylinder and cone.
We can find only B in the given question.

59. The area of the rectangle is 40% more than that of the perimeter of the square. The radius of the cone to the breadth of the rectangle is 1: 2 and the volume of the cone is 308 cm³. The height of the cone is 6 cm and the perimeter of the rectangle is 68 cm. From the statement given in the above question which of the following can be determined.

A. Total surface area of the cone
B. Area of the square

C. If the breadth and length of the rectangle is equal to the breadth and length of the cuboid respectively. Find volume of cuboid?
D. Diagonals of the rectangle

Correct Answer: (b) Only A, B and D
Solution:

Height of the cone = 6 cm
308 = 1/3 × 22/7 × r × r × 6

Radius of the cone = 7 cm
Breadth of the rectangle = 2/1 × 7 = 14 cm

Perimeter of the rectangle = 2(l + b) = 68 cm
L = 34 - 14 = 20 cm

Area of the rectangle = 20 × 14 = 280

Perimeter of the square × (140/100) = 280
Perimeter of the square = 200 cm

Side of the square = 50 cm
Area of the square = 50 × 50 = 2500 cm²

Slanting height of the cone (l) = √(r² + h²)
= √(7² + 6²) = √85

Total surface area of the cone = πr(r + l) = 22/7 × 7 × (7 + √85) = 154 + 22√85 cm²

Diagonal of the rectangle = √(20² + 14²) = √596 cm

c) height of the cuboid is not given.
We can find the answers of A, B and D only.

60. The ratio of the height to radius of the cone is 24: 7 and the slant height is 25 cm. From the statement given in the above question, which of the following can be determined

A. Find Total surface area of the cone
B. If the ratio of the radius of the cylinder to cone is 2: 1, volume of the cylinder is?
C. If the ratio of the breadth of the rectangle to height of the cone is 3: 4 and the perimeter of the rectangle is 76 cm, then length of the rectangle is?
D. What is the sum of the curved surface area of the cone and radius of the cylinder?

Correct Answer: (c) Only C and A
Solution:

25 = √[(24x)² + (7x)²] = 25x
x = 1 cm

Radius of the cone = 7 × 1 = 7 cm
Height of the cone = 24 × 1 = 24 cm

TSA of the cone = πr(r + l) = 22/7 × 7 × (25 + 7)
= 704 cm²

b) Height of the cylinder is not given.
Breadth of the rectangle = 3/4 × 24 = 18 cm
2(1 + 18) = 76

Length of the rectangle = 20 cm

d) Radius of the cylinder is not given.
We can find the answers of A and C.