A cylindrical tub of radius 12 cm contains water to a depth of 20 cm. A spherical iron ball is dropped into the tub and thus the level of water is raised by 6.75 cm. What is the radius of the ball?

From the formula the net change in volume is given as [x + y + z + (xy + yz + zx)/100 + xyz/(100)² ] %
Here, x = 200, y = z = -50 (-ve sign for decrease)
= [ 200 + (-50) + (-50) + (200) x (-50) + (-50) x (-50) + (-50) x (200)/100 + (200) x (-50) x (-50)/(100)² ] %
= 100 + (-10000 + 2500 - 10000)/100 + 500000/100 x 100 ] %
= [ 100 + (-17500)/100 + 50]%
= (150 - 175)% = -25% (-ve sign implies for a decrease )
So, the volume is decreased by 25%.

The net decrease in the surface area of the cube is given by
[x + y + xy/100] %
Here, x = y = 19%
= [-19 + (-19) + (-19) x (-19)/100] %
= [-38 + 361/100] %
= [-38 + 3.61] %= -34.39% (-ve sign implies for decrease)

The decrease in volume is given [ x + y + xy/100 ] %
Here , x = y = -24
= [(-24) + (-24) + (-24) x (-24)/100] %
= [-48 + 576/100] % = -48 + 5.76 = - 42.24 %
(-ve sign denotes decrease in volume)

Let original radius = r and height = h
Original volume = (1/3)?r²h
New radius = 2r and new height = 2h
? New volume = 1/3 x ?(2r)² x 2h = 8/3?r²h
Required ratio = 8/3?r²h : 1/3?r²h = 8 : 1

Volume of Earth dug out = 5 x 4.5 x 2.1 = 47.25 m²
Area over which Each is spread = 13.5 x 25 - 5 x 4.5
= 33.75 - 220 = 11.75 m
Rise in level = 47.25/11.75 = 4.02 m

Volume of the sphere = Volume of the water displaced
Let the required height to which the water rises be h.
Then, ?r₁²h = 4/3 ?r₂³
? 16 h = 4/3 x 27 ? h = 36/16 = 9/4 cm

As the box is painted from inside, the dimensions of the box from inside are
Length = (21 - 05 - 05) = 20 cm
Breadth = (11 - 05 - 05) = 10 cm and height = (6 - 05) = 5.5
Total number of faced to be painted = 4 walls + One base
The dimensions of two of the walls = 2 x (10 x 5.5)
The dimensions of the remaining two = 2 x (20 x 5.5) and that of the base = (20 x 10 ) and so the total area to be pained = 2 x (10 x 5.5) + 2 x (20 x 5.5) + (20 x 10 ) = 530cm³
Since, the total expenses of painting this area is ?70, the rate of painting
= 70/530 = 0.13 = ? 0.1 per cm³ (approx)

Number of boxes = (Volumes of wooden box in cm³) / (Volume of 1 small box)
= (800 x 700 x 600) / (8 x 7 x 6) = 1000000

Let length = 6k, breadth = 5k and height = 4k in cm
? 2(6k x 5k + 5k x 4k + 6k x 4k) = 33300
? 148k² = 33300
? k² = 33300/148 = 225
? k = 15
? Length = 90 cm, Breadth = 75 cm, Height = 60 cm

? ?3a = 8?3
? a = 8
? Surface area = 6a²
= (6 x 8 x 8) cm²
= 384 cm²