Samacheer Kalvi Class 9 Maths Solutions Chapter 7 Mensuration Exercise 7.3

Question 1. Find the volume of a cuboid whose dimensions are
(i) length = 12 cm, breadth = 8 cm, height = 6 cm
(ii) length = 60 m, breadth = 25 m, height = 1.5 m
Answer:
(i) For the first cuboid, we are given the dimensions:
Length \( l = 12 \) cm
Breadth \( b = 8 \) cm
Height \( h = 6 \) cm
The formula for the volume of a cuboid is \( V = l \times b \times h \).
So, Volume \( = (12 \times 8 \times 6) \) cm\(^3\)
Volume \( = 576 \) cm\(^3\). This calculation shows how multiplying the three dimensions gives us the space it occupies.

(ii) For the second cuboid, the dimensions are:
Length \( l = 60 \) m
Breadth \( b = 25 \) m
Height \( h = 1.5 \) m
Using the same volume formula, \( V = l \times b \times h \).
So, Volume \( = 60 \times 25 \times 1.5 \) m\(^3\)
Volume \( = 2250 \) m\(^3\).
In simple words: To find how much space a cuboid takes up, just multiply its length, width, and height together.

🎯 Exam Tip: Always remember to write the correct units (like cm\(^3\) or m\(^3\)) for volume in your final answer.

Question 2. The dimensions of a match box are 6 cm \( \times \) 3.5 cm \( \times \) 2.5 cm. Find the volume of a packet containing 12 such match boxes.
Answer: First, we find the volume of one matchbox.
Length of a match box \( (l) = 6 \) cm
Breadth of a match box \( (b) = 3.5 \) cm
Height of a match box \( (h) = 2.5 \) cm
Volume of one match box \( = l \times b \times h \) cubic units
Volume \( = 6 \times 3.5 \times 2.5 \) cm\(^3\)
Volume \( = 52.5 \) cm\(^3\). This is the space one matchbox takes up.

Next, we find the total volume for 12 match boxes.
Volume of 12 match boxes \( = 12 \times 52.5 \) cm\(^3\)
Volume \( = 630 \) cm\(^3\). Knowing the total volume helps in designing the packet size.
In simple words: First, work out how much space one matchbox takes. Then, multiply that by 12 to find the total space needed for all the matchboxes together.

🎯 Exam Tip: For problems involving multiple identical items, calculate for one item first, then multiply by the total count.

Question 3. The length, breadth and height of a chocolate box are in the ratio 5 : 4 : 3. If its volume is 7500 cm\(^3\), then find its dimensions.
Answer: Let the length of the chocolate box be \( 5x \), the breadth be \( 4x \), and the height be \( 3x \). Ratios help us represent proportional relationships between quantities.
Given that the volume of the chocolate box is \( 7500 \) cm\(^3\).
The formula for volume is \( l \times b \times h = 7500 \).
Substitute the dimensions in terms of \( x \):
\( (5x) \times (4x) \times (3x) = 7500 \)
\( 5 \times 4 \times 3 \times x^3 = 7500 \)
\( 60 x^3 = 7500 \)
Now, we solve for \( x^3 \):
\( x^3 = \frac{7500}{60} \)
\( x^3 = 125 \)
To find \( x \), we take the cube root of 125:
\( x^3 = 5^3 \)
\( x = 5 \)

Now we can find the actual dimensions:
Length of the chocolate box \( = 5 \times 5 = 25 \) cm
Breadth of the chocolate box \( = 4 \times 5 = 20 \) cm
Height of the chocolate box \( = 3 \times 5 = 15 \) cm. These dimensions help in manufacturing the box precisely.
In simple words: When dimensions are given in a ratio, use 'x' to find the real numbers. Multiply the ratio numbers with 'x' to get length, breadth, and height. Then, multiply these to find volume. Solve for 'x' and use it to calculate the true length, breadth, and height.

🎯 Exam Tip: When dealing with ratios, always introduce a common multiplier (like \( x \)) to represent the actual dimensions, then solve for it using the given volume or area.

Question 4. The length, breadth and depth of a pond are 20.5 m, 16 m and 8 m respectively. Find the capacity of the pond in litres.
Answer: First, let's find the volume of the pond in cubic meters.
Length of a pond \( (l) = 20.5 \) m
Breadth of a pond \( (b) = 16 \) m
Depth of a pond \( (h) = 8 \) m
Volume of the pond \( = l \times b \times h \) cubic units
Volume \( = 20.5 \times 16 \times 8 \) m\(^3\)
Volume \( = 2624 \) m\(^3\). This is the space the pond occupies.

Next, we convert the volume from cubic meters to litres. Capacity tells us how much a container can hold, often expressed in litres.
We know that \( 1 \) cubic meter \( (m^3) = 1000 \) litres.
So, the capacity in litres \( = (2624 \times 1000) \) litres
Capacity \( = 2624000 \) litres.
In simple words: Calculate the volume of the pond by multiplying its length, breadth, and depth. Then, since one cubic meter holds 1000 litres, multiply your volume by 1000 to get the total capacity in litres.

🎯 Exam Tip: Always remember the conversion factor for volume: \( 1 \) cubic meter equals \( 1000 \) litres. This is crucial for capacity problems.

Question 5. The dimensions of a brick are 24 cm \( \times \) 12 cm \( \times \) 8 cm. How many such bricks will be required to build a wall of 20 m length, 48 cm breadth and 6 m height?
Answer: First, we need to ensure all measurements are in the same units. We will convert meters to centimeters since the brick dimensions are in centimeters.

For the brick:
Length of a brick \( (l) = 24 \) cm
Breadth of a brick \( (b) = 12 \) cm
Depth (height) of a brick \( (h) = 8 \) cm
Volume of one brick \( = l \times b \times h \)
Volume of one brick \( = 24 \times 12 \times 8 \) cm\(^3\)
Volume of one brick \( = 2304 \) cm\(^3\).

For the wall:
Length of a wall \( (l) = 20 \) m \( = 20 \times 100 = 2000 \) cm
Breadth of a wall \( (b) = 48 \) cm
Height of a wall \( (h) = 6 \) m \( = 6 \times 100 = 600 \) cm
Volume of the wall \( = l \times b \times h \)
Volume of the wall \( = 2000 \times 48 \times 600 \) cm\(^3\)
Volume of the wall \( = 57,600,000 \) cm\(^3\). Using consistent units makes calculations accurate.

Now, to find the number of bricks required, we divide the volume of the wall by the volume of one brick:
Number of bricks \( = \frac{\text{Volume of the wall}}{\text{Volume of one brick}} \)
Number of bricks \( = \frac{2000 \times 48 \times 600}{24 \times 12 \times 8} \)
We can simplify the calculation:
\( = \frac{2000 \times (48/24) \times (600/12)}{8} \)
\( = \frac{2000 \times 2 \times 50}{8} \)
\( = \frac{200000}{8} \)
\( = 25000 \)
Therefore, \( 25000 \) bricks are required to build the wall.
In simple words: First, make sure all measurements are in the same units (like all centimeters). Calculate the total space the wall takes up, and also the space one brick takes up. Then, divide the wall's total space by the brick's space to find out how many bricks are needed.

🎯 Exam Tip: Always convert all dimensions to a consistent unit (e.g., all cm or all m) before performing volume calculations to avoid errors.

Question 6. The volume of a container is 1440 m\(^3\). The length and breadth of the container are 15 m and 8 m respectively. Find its height.
Answer: Let the height of the container be \( h \).
Given:
Length of the container \( (l) = 15 \) m
Breadth of the container \( (b) = 8 \) m
Volume of the container \( = 1440 \) m\(^3\)
The formula for the volume of a cuboid is \( V = l \times b \times h \).
Substitute the given values into the formula:
\( 15 \times 8 \times h = 1440 \)
First, multiply the length and breadth:
\( 120 \times h = 1440 \)
Now, to find \( h \), we divide the volume by the product of length and breadth. This is an example of using inverse operations to find an unknown value.
\( h = \frac{1440}{15 \times 8} \)
\( h = \frac{1440}{120} \)
\( h = 12 \) m
Therefore, the height of the container is \( 12 \) m.
In simple words: If you know the volume, length, and breadth of a container, you can find its height by dividing the volume by (length multiplied by breadth).

🎯 Exam Tip: When a volume is given along with two dimensions, remember that height can be found by dividing the volume by the product of the given length and breadth.

Question 7. Find the volume of a cube each of whose side is
(i) 5 cm
(ii) 3.5 m
(iii) 21 cm
Answer: The formula for the volume of a cube is \( V = a^3 \), where \( a \) is the length of one side. A cube has all its sides of equal length, making its volume calculation straightforward.

(i) For a side length of 5 cm:
Side of a cube \( (a) = 5 \) cm
Volume of a cube \( = a^3 \) cubic units
Volume \( = 5 \times 5 \times 5 \) cm\(^3\)
Volume \( = 125 \) cm\(^3\).

(ii) For a side length of 3.5 m:
Side of a cube \( (a) = 3.5 \) m
Volume of a cube \( = a^3 \) cubic units
Volume \( = 3.5 \times 3.5 \times 3.5 \) m\(^3\)
Volume \( = 42.875 \) m\(^3\).

(iii) For a side length of 21 cm:
Side of a cube \( (a) = 21 \) cm
Volume of a cube \( = a^3 \) cubic units
Volume \( = 21 \times 21 \times 21 \) cm\(^3\)
Volume \( = 9261 \) cm\(^3\).
In simple words: To find the volume of a cube, just multiply the length of one side by itself three times.

🎯 Exam Tip: Remember that for a cube, all dimensions (length, breadth, height) are equal, simplifying the volume formula to side cubed (\( a^3 \)).

Question 8. A cubical milk tank can hold 125000 litres of milk. Find the length of its side in metres.
Answer: First, we need to convert the volume from litres to cubic meters, since we want the side length in meters.
Given volume of the cubical tank \( = 125000 \) litres.
We know that \( 1 \) cubic meter \( (m^3) = 1000 \) litres.
So, to convert litres to cubic meters, we divide by 1000:
Volume in m\(^3\) \( = \frac{125000}{1000} \) m\(^3\)
Volume \( = 125 \) m\(^3\). This conversion is essential for consistent units.

Now, for a cube, the volume is given by \( V = a^3 \), where \( a \) is the length of its side.
So, \( a^3 = 125 \)
To find \( a \), we take the cube root of 125:
\( a^3 = 5^3 \)
\( a = 5 \) m
Therefore, the length of the side of the cubical milk tank is \( 5 \) metres.
In simple words: Change the tank's volume from litres to cubic meters (divide by 1000). Then, find the number that, when multiplied by itself three times, gives this new volume. That number is the length of one side of the tank.

🎯 Exam Tip: When given volume in litres for a cubical container, always convert it to cubic meters first by dividing by 1000 before finding the cube root for the side length.

Question 9. A metallic cube with side 15 cm is melted and formed into a cuboid. If the length and height of the cuboid is 25 cm and 9 cm respectively then find the breadth of the cuboid.
Answer: When a solid is melted and reshaped into another solid, its volume remains the same. This is the principle of conservation of volume.

First, calculate the volume of the original metallic cube:
Side of a cube \( (a) = 15 \) cm
Volume of the cube \( = a^3 \)
Volume of the cube \( = 15 \times 15 \times 15 \) cm\(^3\)
Volume of the cube \( = 3375 \) cm\(^3\).

Now, this volume is equal to the volume of the new cuboid:
Volume of the cuboid \( = 3375 \) cm\(^3\)
For the cuboid, we are given:
Length of a cuboid \( (l) = 25 \) cm
Height of a cuboid \( (h) = 9 \) cm
Let the breadth of the cuboid be \( b \).
The formula for the volume of a cuboid is \( V = l \times b \times h \).
So, \( 25 \times b \times 9 = 3375 \)
\( 225 \times b = 3375 \)
To find \( b \), we divide the volume by the product of length and height:
\( b = \frac{3375}{25 \times 9} \)
\( b = \frac{3375}{225} \)
\( b = 15 \) cm
Therefore, the breadth of the cuboid is \( 15 \) cm. This demonstrates how mass and volume are conserved during physical transformations.
In simple words: When a shape is melted and changed into another shape, its total space (volume) stays the same. Calculate the volume of the first shape (the cube). Then, use that volume and the given parts of the new shape (the cuboid) to find its missing dimension.

🎯 Exam Tip: For problems where a solid is melted and recast, remember that the volume of the original solid is equal to the volume of the new solid formed.