BANK & INSURANCE (MENSURATION) PART 2

Let side of square be s and side of equilateral triangle be a
=> √2s = a
=> s = a/√2

Area of square = s² = 18√3
=> a²/2 = 18√3

∴ Area of equilateral triangle = √3 a²/4 = (√3/2) × 18√3 = 54/2 = 27 sq. cm

Given:
A rhombus has one of its diagonal 65% of the other.
A square is drawn using the longer diagonal as a side.

Concept used:
Area of rhombus = ½ (diagonals product)
Area of square = side × side

Calculations:
Let diagonal (larger) of rhombus be 100 cm

Let the diagonal (smaller) diagonal be 65 cm (65% of larger diagonal)
Area of Rhombus = ½ (100 × 65) = 3250
Side of square = 100 cm (equal to larger diagonal)
Area of square = (100 × 100) = 10000
Ratio,
⇒ Rhombus : Square = 3250 : 10000 = 13 : 40

Area of a Square = Area of Rectangle
Length of Rectangle = Side of Square + 5
Breadth of Rectangle = Side of Square − 4

Formula Used:
Area of Square = (Side of Square)²
Area of Rectangle = Length × Breadth
Perimeter of Rectangle = 2 × (Length + Breadth)

Calculation:
Let a be the side of the Square, l be the length of the Rectangle and b be the breadth of the Rectangle
a² = l × b
l = a + 5
b = a − 4

⇒ a² = (a + 5) × (a − 4)
⇒ a² = a² + 5a − 4a − 20
⇒ a = 20

l = 20 + 5 = 25
b = 20 − 4 = 16

now, Perimeter of the Rectangle = 2 × (25 + 16)
= 82 cm

Area of the square = 1521 sq. m.
∴ Side of the square = √1521 = 39 m.
∴ Breadth of the rectangle = 1/3 of 39 = 13 m.
∴ Length of the rectangle = 2 × 13 = 26 m
∴ Area of the rectangle = 13 × 26 = 338 sq. m.
∴ Difference between the area of the square and the area of the rectangle,
⇒ 1521 − 338 = 1183 sq. m.

Area of rectangular field = 1260
⇒ x(x − 12) = 1260
⇒ x² − 12x − 1260 = 0
⇒ x² − 42x + 30x − 1260 = 0
⇒ (x − 42)(x + 30) = 0
⇒ x = 42

∴ Length of rectangular field = 42 m
Breadth of rectangular field = 42 − 12 = 30 m

Perimeter of semi circular garden = Perimeter of rectangular field = 2(30 + 42)
⇒ 2 × 72
⇒ 144 m

So, the cost of fencing the semi-circular garden = 144 × 15 = Rs. 2160

Let the ratio be x
So, length = 5x and breadth = 4x
According to question,

a³ = 216
⇒ a = 6 cm
So, height of cuboid = 6 cm

And
Volume of cuboid = l × b × h
l × b × h = 480
⇒ 5x × 4x × 6 = 480
⇒ 120x² = 480
⇒ x² = 4

So, x = +2 or x = −2
So, length of cuboid = 5 × 2 = 10 cm
Breadth of cuboid = 4 × 2 = 8 cm

Now,
Diagonal of cube = 6√3
Diagonal of cuboid = √(10² + 8² + 6²) = 10√2

Hence, required ratio = 6√3 : 10√2

Let side of a square be a
Let Length and Breadth of a rectangle be l and b respectively

a² = l × b
⇒ a² = [(a + 8) × (a − 6)]
⇒ a² = a² − 6a + 8a − 48
⇒ 2a = 48 cm
⇒ a = 24 cm

⇒ l = (a + 8) = (24 + 8) = 32 cm
⇒ b = (a − 6) = (24 − 6) = 18 cm

Perimeter of rectangle = 2(32 + 18) cm = 100 cm
∴ Perimeter of rectangle is 100 cm

Side × √2 = 29√2
Side = 29 cm
Perimeter of a square = 4 × side = 4 × 29 = 116 cm

According to the question,
Perimeter of the square = Perimeter of the rectangle
Perimeter of rectangle = 2(L + B) = 116
⇒ L + B = 58

⇒ L + (L − 8) = 58 (Given length of rectangle is 8 cm more than the breadth)
⇒ L = 33 cm and B = 25 cm

Area of the rectangle = L × B = 33 × 25 = 825 cm²

Let the required radius of the cylinder be r
We can calculate the radius by using the relation for the curved surface area:
176 = 2 × π × r × 14
⇒ r = 2 cm

When the radius is doubled, the new radius = 2 × 2 = 4 cm

The new volume of the cylinder = π × 42 × 14 = 704 cm³