Surface Areas and Volumes: 9 Math

NCERT Exercise 13.5 part-2: 9th math

Surface Areas And Volumes Class nine Math NCERT Exercise 13.5  Question (5) The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank, if its length and depth are respectively 2.5 m and 10 m.

Solution

Given, Capacity of the tank = 50000 Litre

= 50000/1000

(∵ 1 m³= 1000 `l`)

⇒ Capacity of the tank = 50 m³

And, Length of the tank (`l`) = 2.5 m

And, Height (depth) of the tank (h) = 10 m

Therefore, Breadth of the given tank (b) = ?

We know that, volume of a cuboid `=lxxbxxh`

Therefore, volume of the given tank = 2.5 m × b × 10 m

⇒ 50 m³ = 25 m² × b

`=> b =(50 m^3)/(25m^2)`

⇒ Breadth (b) = 2 m

Thus, breadth of the given tank = 2 m Answer

Surface Areas And Volumes Class nine Math NCERT Exercise 13.5  Question (6) A village, having a population of 4000, requires 150 litres of water per head per day. It has a tank measuring 20 m × 15 m × 6m. For how many days will the water of this tank last?

Solution

Given, Population = 4000

Consumption of water per head per day = 150 `l`

And, Length of the water tank (`l`) = 20 m

And, Breadth of the water tank (b) = 15 m

And, Height of the water tank (h) = 6 m

Thus, number of days for which water is sufficient in the given tank = ?

Now, we know that, Volume of a cuboid `=lxxbxxh`

Therefore, volume of the given cuboidal tank =

= 6 m × 15 m × 20 m

= 90 m² × 20 m

= 1800 m³

Thus, volume of the tank = 1800 m³

Calculation of water sufficient for how many days

Now, we know that, 1 m³ = 1000 `l`

Therefore, 1800 m³ = 1800000 `l`

And, since, as per question, consumption of water per head per day = 150 `l` and population of the village is equal to 4000

Therefore, in 1 day consumption of water for whole population = 4000 × 150 `l`

Thus, per day consumption of water = 600000 `l`

Now volume of water in the tank = 1800000 `l`

Now, since 600000 litres of water last for 1 day

Therefore, 1 litre of water last for `1/600000` days

Therefore, 1800000 litres of water last for `1/600000xx1800000` days

= 3 days

Thus, water in the tank is sufficient for 3 days Answer

Surface Areas And Volumes Class nine Math NCERT Exercise 13.5  Question (7) A godown measures 40 m × 25 m × 15 m. Find the maximum number of wooden crates each measuring 1.5 m × 1.25 m × 0.5 m that can be stored in the godown.

Solution

Given, Length of the godown (`l`) = 40 m,

Bredath of the godown (b) = 25 m,

Height of the godown (h) = 15 m,

And, Length of the wooden crates (`l_c`) = 1.5 m,

Breadth of the wooden crates (b_c) = 1.25 m,

And, Height of the wooden crates (h_c) = 0.5 m

Therefore number of given wooden crates can be stored in the given godown = ?

Now, we know that, Volume of a cuboid `=lxxbxxh`

Therefore, Volume of the given godown

= 40 m × 25 m × 15 m

= 1000 m² × 15 m

= 15000 m³

Thus, volume of the godown = 15000 m³

And, Volume of the crate

= 1.5 m × 1.25 m × 0.5 m

= 1.875 m² × 0.5 m

= 0.9375 m³

Thus, volume of the crate = 0.9375 m³

Now, number of crates that can be stored in the godown

= volume of godown/volume of crates

`=(15000 m^3)/(0.9375m^3)`

= 16000 crated

Thus, number of crates which can be stored in the given godown = 16000 Answer

Alternatively, number of crates can be calculated as follows

Now, number of crates that can be stored in the godown

= volume of godown/volume of crates

`=(40mxx25mxx15m)/(1.5mxx1.25mxx0.5m)`

= 16000 crates

Thus, number of crates which can be stored in the given godown = 16000 Answer

Surface Areas And Volumes Class nine Math NCERT Exercise 13.5  Question (8) A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.

Solution

Given, Side of the solid cube = 12 cm

And, number of equal volumes of cubes cut from the given solid cube = 8

Therefore, side of the new cube = ?

And, ratio between surface area of old cube and new cube = ?

Now, we know that, Volume of cube = (side)³

Therefore, volume of the given cube = (12 cm)³

= 1728 cm³

And, Number of small cubes of equal volumes cut from the given solid cube = 8

Therefore, volume of small cube = 1728/8

= 216 cm³

Thus, volume of one small cube = 216 cm³

Now, let edge of small cube = s

Thus, volume of a small cube = (s)³

⇒ 216 cm³ = s³

`=>s = root3 (216 cm^3)`

⇒ s = 6 cm

Thus, side of one small cube = 6 cm

Calculation of ratio of surface area of given two cubes

Now, we know that, surface area of a cube = 6 × (side)²

Therefore, surface area of given solid cube = 6 × (12 cm)²

⇒ surface area of given solid cube = 6 × 12 cm × 12 cm

And, surface area of new small cube = 6 × (6 cm)²

Thus, surface area of new small cube = 6 × 6 cm × 6 cm

Now, ratio of surface area of solid cube and surface area of new small cube

`=(6xx12cmxx12cm)/(6xx6cmxx6cm)`

`= (2xx2)/1`

= 4 : 1

Thus, ratio of surface area of solid cube and new small cube = 4:1

Thus, side of new small cube = 6 cm and ratio of solid cube and new small cube = 4:1 Answer

Surface Areas And Volumes Class nine Math NCERT Exercise 13.5  Question (9) A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute?

Solution

Given, Depth of the river = 3 m,

Breadth of the river = 40 m

And, rate of flowing of water = 2 km/hour

This means length of the surface of water = 2 km

= 2 × 1000 = 2000 m

⇒ Length of the surface of water = 2000 m

Therefore, fall of water into the sea in 1 minute = ?

Now, since length of surface of water flow into the river in 1 hour (60 minute) = 2000 m

Therefore, length of surface of water flow into the river in 1 minute = 2000 m/60

= 100/3 m

Thus, flow of water into the river in 1 minute = 100/3 meter

Thus, length of the surface of water = 100/3 m

Now, we know that, Volume of a cuboid `=lxxbxxh`

Therefore, volume of water flow into the sea in 1 minute `=100/3mxx40 mxx3m`

= 100 m × 40 m × 1 m

= 4000 m³

Thus, volume of water of the given falls into the sea in 1 minute = 4000 m³ Answer

Alternate method

Given, Depth of the river = 3 m,

Breadth of the river = 40 m

And, rate of flowing of water = 2 km/hour

This means length of the surface of water = 2 km

= 2 × 1000 = 2000 m

⇒ Length of the surface of water = 2000 m

Therefore, fall of water into the sea in 1 minute = ?

Now, we know that, Volume of a cuboid `=lxxbxxh`

Thus, volume of water falls into river in 1 hour

= 2000 m × 40 m × 3 m

= 80000 m² × 3 m

= 240000 m³

Thus, volume of water flows into the sea in 60 minutre (1 hour) = 240000 m³

Therefore, volume of water flows into the sea in 1 minute `=(240000m^3)/60`

= 4000 m³

Thus, volume of water of the given falls into the sea in 1 minute = 4000 m³ Answer

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