Chapter 11: Surface Areas and Volumes

Dimensions of a matchbox (a cuboid) are l×b×h = 4 cm×2.5 cm×1.5 cm, respectively

Formula to find the volume of matchbox = l×b×h = (4×2.5×1.5) = 15

Volume of matchbox = 15 cm³

Now, volume of 12 such matchboxes = (15×12) cm³ = 180 cm³

Therefore, the volume of 12 matchboxes is 180 cm³.

Dimensions of a cuboidal water tank are l = 6 m and b = 5 m, and h = 4.5 m

Formula to find the volume of the tank, V = l×b×h

Putting the values, we get

V = (6×5×4.5) = 135

The volume of the water tank is 135 m³

Again,

We are given that the amount of water that 1m³ volume can hold = 1000 l

Amount of water, 135 m³volume hold = (135×1000) litres = 135000 litres

Therefore, given cuboidal water tank can hold up to 135000 litres of water.

Given:

Length of the cuboidal vessel, l = 10 m

Width of the cuboidal vessel, b = 8m

Volume of the cuboidal vessel, V = 380 m³

Let the height of the given vessel be h.

Formula for volume of a cuboid, V = l×b×h

Using the formula, we have

l×b×h = 380

10×8×h= 380

Or h = 4.75

Therefore, the height of the vessels is 4.75 m.

The given pit has its length(l) as 8m, width (b)as 6m and depth (h)as 3 m.

Volume of cuboidal pit = l×b×h = (8×6×3) = 144 (using formula)

The required Volume is 144 m³

Now,

Cost of digging per m³ volume = Rs 30

Cost of digging 144 m³ volume = Rs (144×30) = Rs 4320

The length (l) and depth (h) of the tank are 2.5 m and 10 m, respectively.

To find: The value of breadth, say b.

Formula to find the volume of a tank = l×b×h = (2.5× b×10) m³= 25b m³

The capacity of tank= 25b m³, which is equal to 25000b litres

Also, the capacity of a cuboidal tank is 50000 litres of water (Given).

Therefore, 25000 b = 50000

This implies that b = 2

Therefore, the breadth of the tank is 2 m.

Length of the tank = l = 20 m

Breadth of the tank = b = 15 m

Height of the tank = h = 6 m

Total population of a village = 4000

Consumption of water per head per day = 150 litres

Water consumed by the people in 1 day = (4000×150) litres = 600000 litres …(1)

Formula to find the capacity of the tank, C = l×b×h

Using the given data, we have

C = (20×15×6) m³= 1800 m³

Or C = 1800000 litres

Let water in this tank last for d days.

Water consumed by all people in d days = Capacity of the tank (using equation (1))

600000 d =1800000

d = 3

Therefore, the water in this tank will last for 3 days.

From the statement, we have

Length of the godown = 40 m

Breadth = 25 m

Height = 15 m

Whereas,

Length of the wooden crate = 1.5 m

Breadth = 1.25 m

Height = 0.5 m

The godown and wooden crate are in cuboidal shape. Find the volume of each using the formula V = lbh

Now,

Volume of godown = (40×25×15) m³ = 15000 m³

Volume of a wooden crate = (1.5×1.25×0.5) m³ = 0.9375 m³

Let us consider that, n wooden crates can be stored in the godown, then

Volume of n wooden crates = Volume of godown

0.9375×n =15000

Or n= 15000/0.9375 = 16000

Hence, the number of wooden crates that can be stored in the godown is 16,000.

Side of a cube = 12cm (Given)

Find the volume of the cube.

Volume of cube = (Side)³ = (12)³cm³= 1728cm³

Surface area of a cube with side 12 cm = 6a² = 6(12) ² cm² …(1)

The cube is cut into eight small cubes of equal volume; say, the side of each cube is p.

Volume of a small cube = p³

Surface area = 6p² …(2)

Volume of each small cube = (1728/8) cm³ = 216 cm³

Or (p)³ = 216 cm³

Or p = 6 cm

Now, Surface areas of the cubes ratios = (Surface area of the bigger cube)/(Surface area of the smaller cubes)

From equations (1) and (2), we get

Surface areas of the cubes ratios = (6a²)/(6p²) = a²/p² = 12²/6² = 4

Therefore, the required ratio is 4 : 1.

Given:

Depth of river, h = 3 m

Width of river, b = 40 m

Rate of water flow = 2km per hour = 2000m/60min = 100/3 m/min

Now, volume of water flowed in 1 min = (100/3) × 40 × 3 = 4000m³

Therefore, 4000 m³ water will fall into the sea in a minute.