RD Sharma 2020 solution class 9 chapter 18 Surface Area and Volume of a Cuboid and Cube Exercise 18.2

Exercise 18.2

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Question 1:

A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of water can it hold?

Answer 1:

Dimensions of the water tank, l = 6 m, b = 5 m, h = 4.5 m

We need to find the capacity of the tank

Capacity of the tank,

The tank can hold of water.

Question 2:

A cubical vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid?

Answer 2:

We have,

Length of the vessel

Width of the vessel

Capacity of the vessel

Let: Minimum required height of the vessel

So,

Thus, to hold of liquid, the vessel must be minimum high.

Question 3:

Find the cost of digging a cuboidal pit 8 m long, 6 m broad and 3 m deep at the rate of Rs 30 per m³.

Answer 3:

We have, the dimensions of the cubical pit are,

Length

Breadth

Depth

Rate of digging

Volume of the pit,

The cost of digging,

The cost of digging the pit is.

Question 4:

If the areas of three adjacent faces of a cuboid are 8 cm², 18 cm³ and 25 cm³. Find the volume of the cuboid.

Answer 4:

We know that, areas of three adjacent faces of the cuboid are respectively.

Where,

Length of the cuboid

Breadth of the cuboid

Height of the cuboid

Let,

Volume of the cuboid

We have, areas of three adjacent faces of the cuboid are respectively,

So their product,

The volume of the cuboid is.

Question 5:

The breadth of a room is twice its height, one half of its length and the volume of the room is 512 cu. dm. Find its dimensions.

Answer 5:

Let,

Length of the room

Breadth of the room

Height of the room

Volume of the room

We have, b = 2h, l = 2b and volume of room is 512 dm³

We have to find the dimensions

We know that

We have,

Therefore,

Hence, the dimensions of the cuboid are,

Length, Breadth, Height
Therefore, length is equal to 16 dm, breadth is 8 m  and Height is 4 dm.

Question 6:

Three metal cubes with edges 6 cm, 8 cm and 10 cm respectively are melted together and formed into a single cube. Find the volume, surface area and diagonal of the new cube.

Answer 6:

Let,

Sides of the three small cubes

Volumes of the three small cubes

Side of the new cube formed

Volume of the new cube formed

Surface area of the new cube formed

Diagonal of the new cube formed

We have,

We need to find the volume, surface area and diagonal of the new cube

Now,

We know,

So, the volume, surface area and the diagonal of the new cube will be respectively.

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Question 7:

Two cubes, each of volume 512 cm³ are joined end to end. Find the surface area of the resulting cuboid.

Answer 7:

We have volume of each cube

Let,

Side of each cube

The two cubes are joined together and we are asked to find the surface area of new cuboid

We know that,

When the two cubes are joined end to end,

Dimensions of the resulting cuboid are,

Length

Breadth

Height

Hence, its surface area

The surface area of the resulting cuboid will be .

Question 8:

A metal cube of edge 12 cm is melted and formed into three smaller cubes. If the edge of the two smaller cubes are 6 cm and 8 cm, find the edge of the third smaller cube.

Answer 8:

Let,

Side of the cube

Volume of the cube

Sides of the three smaller cubes

Volumes of the three smaller cubes

We have,

,

We know that,

Edge of the third smaller cube is .

Question 9:

The dimensions of a cinema hall are 100 m, 50 m and 18 m. How many persons can sit in the hall, if each person requires 150 m³ of air?

Answer 9:

Dimensions of the cinema hall are,

Length

Breath

Height

Each person requires of air (say, v)

We are asked to find the number of persons who can sit in the cinema hall

Let,

Volume of the hall, then

The number of people that can sit in the hall,

Maximum people can sit in the hall.

Question 10:

Given that 1 cubic cm of marble weighs 0.25 kg, the weight of marble block 28 cm in with and 5 cm thick is 112 kg. Find the length of the block.

Answer 10:

We are given that 1 cubic cm of marble weighs 0.25 kg

Let,

Volume of the block

Length of the block

We have,

Width of the block

Thickness of the block

Weight of the block

We need to find the length of the block

We have, of marble occupies of volume.

So, of marble will occupy the volume,

The length of the block is.

Question 11:

A box with lid is made of 2 cm thick wood. Its external length, breadth and height are 25 cm, 18 cm and 15 cm respectively. How much cubic cm of a liquid can be placed in it?  Also, find the volume of the wood used in it.

Answer 11:

External dimensions of the box are,

Length

Breadth

Height

Thickness of the wood

We need to find the volume used

So, internal dimensions of the box are,

Length

Breadth

Height

Capacity of the box,

Volume of the wood,

Maximum of liquid can be placed in the box.

Volume of the wood used in the box is .

Question 12:

The external dimensions of a closed wooden box are 48 cm, 36 cm, 30 cm. The box is made of 1.5 cm thick wood. How many bricks of size 6 cm × 3 cm × 0.75 cm can be put in this box?

Answer 12:

External dimensions of the closed wooden box,

Length

Breath

Height

Thickness of the wood

We need to find number of bricks that can be put inside the box of dimension

Internal dimensions of the box,

Length

Breadth

Height

Capacity of the box,

Volume of each brick,

Number of bricks that can be put in the box,

The box can contain maximum bricks.

Question 13:

A cube of 9 cm edge is immersed completely in a rectangular vessel containing water. If the dimensions of the base are 15 cm and 12 cm, find the rise in water level in the vessel.

Answer 13:

“When an object is completely immersed in a liquid, the volume of the liquid displaced is equal to the volume of the object”

Using this principle, we shall now solve this problem.

We have,

Edge of the immersed cube

Length of the rectangular container

Breadth of the rectangular container

Let,

The rise in water level

Volume of the immersed cube

As per the above mentioned principle,

The rise in water level is .

Question 14:

A field is 200 m long and 150 m broad. There is a plot, 50 m long and 40 m broad, near the field. The plot is dug 7 m deep and the earth taken out is spread evenly on the field. By how many metres is the level of the field raised? Give the answer to the second place of decimal.

Answer 14:

We have, dimensions of the plot that is dug,

Length

Breadth

Depth

Length of the field

Breadth of the field

We need to find the level of field raised

Here, the volume of earth taken out,

So the rise in the level of the field

The level of the field is raised by .

Question 15:

A field is in the form of a rectangle of length 18 m and width 15 m. A pit, 7.5 m long, 6 m broad and 0.8 m deep, is dug in a corner of the field and the earth taken out is spread over the remaining area of the field. Find out the extent to which the level of the field has been raised.

Answer 15:

We have,

Length of the field (L) = 18 m

Width of the field (B) = 15 m

Length of the pit (l) = 7.5 m

Breadth of the pit (b) = 6 m

Depth of the pit (h) = 0.8 m

We have to find the level of field raised

Volume of the earth dug out

The area on which the earth has to be spread,

The rise in the level of the field

The level of the field has been raised to .

Question 16:

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 × 6 m. For how many days will the water of this tank last?

Answer 16:

Let,

Number of days the water will last for

One villager requires of water per day.

Population of the village is 4000

Measurement of the tank is

We need to find for how many days the water of the tank will last

The requirement of water for all 4000 villagers for days,

But we are given;

The water of this tank will last for.

Question 17:

A child playing with building blocks, which are of the shape of the cubes, has built a structure as shown in the given figure. If the edge of each cube is 3 cm, find the volume of the structure built by the child.
 

Answer 17:

We have,

Number of boxes

In the above structure we need to find the total volume

Edge of each cube

Volume of each cube

Hence, total volume of the structure,

The volume of the structure built by the child is.

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Question 18:

A godown measure 40 m × 25 m × 10 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.

Answer 18:

We have,

Volume of the godown

Volume of each crate

We need to find the maximum number of crates in the godown that can be placed

Hence, the number of crates that can be stored,

But, we can not place this amount of crates in the godown, as this is not an integer.

So, we can place maximumcrates in the godown.

Question 19:

A wall of length 10 m was to be built across an open ground. The height of the wall is 4 m and thickness of the wall is 24 cm. If this wall is to be built up with bricks whose dimensions are 24 cm × 12cm × 8cm, how many bricks would be required.

Answer 19:

We have,

Length of the wall

Height of the wall

Thickness of the wall

Dimension of the brick is

We need to find the number of bricks

Here,

Volume of the wall,

Dimensions of the brick are,

So, number of bricks in the wall,

As this is not an integer, we should take least integer greater than.

So, we need bricks to build the wall.

Question 20:

If V is the volume of a cuboid of dimensions a, b, c and S is its surface area, then prove that

$\frac{1}{V}=\frac{2}{S} \left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

Answer 20:

We have,

Volume of the cuboid

Surface area of the cuboid

Dimensions of the cuboid

We need to prove,

We know that,

And

Hence,

Question 21:

The areas of three adjacent faces of a cuboid are x, y and z. If the volume is V, prove that V² = xyz.

Answer 21:

Let,

Length of the cuboid

Breadth of the cuboid

Height of the cuboid

Volume of the cuboid

Areas of three adjacent faces of the cuboid

We know that, areas of three adjacent faces of the cuboid are lb, bh, and hl respectively

Hence,

Hence,

Question 22:

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?

Answer 22:

River is flowing at the speed of,

Width of the river

Depth of the river

We need to find the water flowed in one minute

So, in one minute, the river covers the distance (say, l) of .

Quantity of water that will fall into the sea in one minute,

So, in one minute of water will fall into the sea.

Question 23:

Water in a canal 30 dm wide and 12 dm deep, is flowing with a velocity of 100 km per hour. How much area will it irrigate in 30 minutes if 8 cm of standing water is desired?

Answer 23:

We are given;

Velocity of the water

So, in 30 min, it will go the distance (say, l) 5,00,000 dm.

Width of the canal

Depth of the canal

In 30 min, quantity of water flown,

If 8 cm of standing water is desired, then the area that will be irrigated,

In 30 min, it will irrigate the area of .

Question 24:

Half cubic metre of gold-sheet is extended by hammering so as to cover an area of 1 hectare. Find the thickness of the gold-sheet.

Answer 24:

Volume of the gold sheet,

Area it covers,

Let,

Thickness of the sheet

We know that,

Thickness of the sheet is .

Question 25:

How many cubic centimetres of iron are there in an open box whose external dimensions are 36 cm, 25 cm and 16.5 cm, the iron being 1.5 cm thick throughout? If 1 cubic cm of iron weighs 15 g, find the weight of the empty box in kg.

Answer 25:

We have,

External dimensions of the iron box are,

Length

Breadth

Height

Thickness of ironand of iron weighs

We are asked to find the volume of the metal used in the box and weight of the empty box

Internal dimensions of the box,

Length

Breadth

Height

Let,

External volume of the box

Internal volume of the box

Volume of the iron

So,

We have, of iron weighs,

So, weight of of iron,

In that open box, there areof iron, and weight of the empty box is .

Question 26:

A rectangular container, whose base is a square of side 5 cm, stands on a horizontal table, and holds water upto 1 cm from the top. When a cube is placed in the water it is completely submerged, the water rises to the top and 2 cubic cm of water overflows. Calculate the volume of the cube and also the length of its edge.

Answer 26:

“When an object is completely immersed in a liquid, the volume of the liquid displaced is equal to the volume of the object”

Using this principle, we shall now solve this problem.

We have,

Length of the container

Breadth of the container

Height to which the water raised

Volume of the water displaced,

= volume of the water raised + volume of the water over flown

We need to calculate the volume and edge of the cube

Let,

Volume of the cube submerged

Edge of the cube submerged

According to the principle mentioned above,

Volume of the cube is and edge of the cube is .

Question 27:

A rectangular tank is 80 m long and 25 m broad. Water flows into it through a pipe whose cross-section is 25 cm², at the rate of 16 km per hour. How much the level of the water rises in the tank in 45 minutes.

Answer 27:

It is given that

Length of the tank (l) = 80 m

Breadth of the tank (b) = 25 m

= 2500 cm

Area of the cross section of the pipe is 25 cm²

The water flows at the rate of 16 km/hr.

We are asked to find the level of tank raised in 45 minutes

In 45 minutes, the water through the pipe will go,

Area of the cross-section of the pipe is 25 cm².

So, the quantity of the water poured in 45 minutes,

Let,

Height to which the water is raised

So,

In 45 minutes, the water level rises by .

Question 28:

Water in a rectangular reservoir having base 80 m by 60 m is 6.5 deep. In what time can the water be emptied by a pipe of which the cross-section is a square of side 20 cm, if the water runs through the pipe at the rate of 15 km/hr.

Answer 28:

Dimensions of the reservoir are,

Length

Breadth

Depth

Side of the cross-section of the pipe

The flow of water through the water is;

We are asked to find the time in which the reservoir can be emptied

Here, volume of the water in the reservoir,

Since the side of the cross-section of the pipe

So, the area of the cross-section of the pipe,

Velocity of the water,

Let,

Time required emptying the reservoir

So,

Using that pipe, the water is emptied in.