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9.6 - Volume and Capacity

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Understanding Volume and Capacity

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Teacher
Teacher

Today, we'll explore two important concepts: volume and capacity. Can anyone tell me what volume means?

Student 1
Student 1

Isn't volume how much space an object takes up?

Teacher
Teacher

Exactly! Volume is the space occupied by a solid. Now, what about capacity?

Student 2
Student 2

Capacity is how much a container can hold, right?

Teacher
Teacher

Correct! So we have volume for solid objects and capacity for containers. Remember, volume measures space while capacity measures contents.

Student 3
Student 3

Can you give us a practical example?

Teacher
Teacher

Sure! If a tin holds 100 cm³ of water, that means its capacity is also 100 cm³.

Teacher
Teacher

So, to remember: Volume = Space, Capacity = Contents. Which aids in understanding the difference better!

Unit Relationships

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Teacher
Teacher

Now let's dive into unit relationships. Who can tell me how many cubic centimeters are in a liter?

Student 4
Student 4

I think it's 1000 cubic centimeters in a liter.

Teacher
Teacher

That's right! And what about milliliters?

Student 1
Student 1

1 mL equals 1 cm³!

Teacher
Teacher

Perfect! So from this, we see the connection: 1 L = 1000 cm³ means 1 m³ = 1,000,000 cm³ = 1000 L.

Student 2
Student 2

That sounds a lot!

Teacher
Teacher

It is! Remember, each relation helps us convert and understand volumes in practical situations.

Practical Application

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Teacher
Teacher

Let's consider real-life applications! In Example 8, if we need to find the height of a cuboid with a volume of 275 cm³ and a base area of 25 cm², how would we approach this?

Student 3
Student 3

We can use the formula: Height = Volume / Base Area!

Teacher
Teacher

Exactly! That gives us a height of 11 cm. Good job! Now, in Example 9, how can we find out how many boxes can fit in a godown?

Student 4
Student 4

We need the volume of the godown and divide it by the volume of a box!

Teacher
Teacher

Correct again! The godown's volume is 72000 m³, and if each box holds 0.8 m³, we find it fits 90,000 boxes.

Student 1
Student 1

That’s a lot of boxes!

Teacher
Teacher

Indeed! To remember, volume helps us understand how much we can store, while capacity tells how much a container can hold.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section explores the concepts of volume and capacity, highlighting their definitions and the relationship between them.

Standard

In this section, volume refers to the space occupied by an object, while capacity refers to the amount a container can hold. The relationship between cubic centimeters and liters is established, providing context for practical applications in measurement.

Detailed

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Audio Book

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Understanding Volume vs. Capacity

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There is not much difference between these two words.

(a) Volume refers to the amount of space occupied by an object.

(b) Capacity refers to the quantity that a container holds.

Detailed Explanation

In this chunk, we differentiate between two related concepts: volume and capacity. Volume is about the space that an object takes up, whereas capacity refers to how much liquid or substance a container can hold. For example, if you have a box filled with toys, the volume is the space the box occupies, while the capacity is how many toys can fit inside the box.

Examples & Analogies

Think of a big balloon filled with air. The volume of the balloon is how much space the air takes inside it, while the capacity would be how much air the balloon could hold if it were empty.

Measurement Conversion

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Note: If a water tin holds 100 cm3 of water then the capacity of the water tin is 100 cm3.

Capacity is also measured in terms of litres. The relation between litre and cm3 is,
1 mL = 1 cm3, 1 L = 1000 cm3. Thus, 1 m3 = 1000000 cm3 = 1000 L.

Detailed Explanation

This chunk provides important relationships between different units of measurement. For example, a small amount of liquid measured in millilitres (mL) is equivalent to cubic centimeters (cm3). Also, 1 litre is equal to 1000 cubic centimeters, and this gives us a straightforward path to convert larger volumes of liquid into more manageable units.

Examples & Analogies

Imagine filling a bottle with water. If you know that 1 litre of water is the same as 1000 cm3, then if you fill your water bottle with 500 cm3 of water, you can say that you have filled it with half a litre.

Example on Finding Height of a Cuboid

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Example 8: Find the height of a cuboid whose volume is 275 cm3 and base area is 25 cm2.

Solution: Volume of a cuboid = Base area × Height

Hence height of the cuboid = Volume/Base area = 275/25 = 11 cm

Height of the cuboid is 11 cm.

Detailed Explanation

This example illustrates how you can find the height of a cuboid if you know its volume and the area of its base. The formula used here states that the volume is the product of the base area and the height. By rearranging this formula, we can solve for height, making it easy to find what we're looking for.

Examples & Analogies

Imagine a shoebox filled with shoes. If you know how much space all the shoes take together (the volume), and you already know the area of the base of the shoebox, you can figure out how tall the shoebox is. In this case, finding the height is just like figuring out how deep the shoes are packed into the box.

Example on Storing Boxes in a Godown

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Example 9: A godown is in the form of a cuboid of measures 60 m × 40 m × 30 m. How many cuboidal boxes can be stored in it if the volume of one box is 0.8 m3?

Solution: Volume of one box = 0.8 m3
Volume of godown = 60 × 40 × 30 = 72000 m3

Volume of the godown / Volume of one box = 72000 / 0.8 = 90000

Hence, the number of cuboidal boxes that can be stored in the godown is 90,000.

Detailed Explanation

In this example, we calculate how many smaller boxes can fit into a larger space, or godown, which is a cuboidal structure. First, we find the total volume of the godown by multiplying its dimensions. Then, we divide this volume by the volume of one smaller box to get the total number of boxes that can fit inside.

Examples & Analogies

Picture a large warehouse storing small boxes of products. If the warehouse can store a total of 72,000 m3 of product and each box takes up 0.8 m3, you can easily calculate how many boxes can fit by simply dividing the total warehouse capacity by the capacity of one box. It's like figuring out how many jars of jam can fit in a pantry!

Example on Volume of a Cylinder

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Example 10: A rectangular paper of width 14 cm is rolled along its width and a cylinder of radius 20 cm is formed. Find the volume of the cylinder.

Solution: Height of the cylinder = h = 14 cm
Radius = r = 20 cm

Volume of the cylinder = V = π r2 h = (22/7) × 20 × 20 × 14 = 17600 cm3.

Hence, the volume of the cylinder is 17600 cm3.

Detailed Explanation

This example explores how to compute the volume of a cylinder formed by rolling a piece of rectangular paper. The width of the paper becomes the cylinder's height, and the radius is given. By using the formula for the volume of a cylinder, we substitute the known values to find the volume.

Examples & Analogies

Think of wrapping a gift with a piece of wrapping paper. When you roll the wrapping paper into a cylindrical shape around the gift, the height of the cylinder is how wide the paper is, and the radius is how round the cylinder is. You can calculate how much space the cylindrical gift takes up just as you would for any other object!

Example on Volume of a Folded Cylinder

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Example 11: A rectangular piece of paper 11 cm × 4 cm is folded without overlapping to make a cylinder of height 4 cm. Find the volume of the cylinder.

Solution: Length of the paper becomes the perimeter of the base of the cylinder and width becomes height.
Let radius of the cylinder = r and height = h

Perimeter of the base of the cylinder = 2πr = 11
or 2πr = 11
Therefore, r = 11/(2π) m.

Volume of the cylinder = V = πr2h.
Hence, the volume of the cylinder is 38.5 cm3.

Detailed Explanation

In this example, a rectangular paper is utilized to create a cylinder. The perimeter of the base of the cylinder is formulated using the length of the paper. Knowing the height allows the calculation of the volume by substituting the radius found from the perimeter back into the cylinder volume formula.

Examples & Analogies

Imagine rolling up a piece of parchment paper to create a scroll. The way the parchment is folded gives it both height and circularity, similar to how you can calculate its volume based on how tightly it's rolled.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Volume: Refers to the amount of space an object occupies.

  • Capacity: Refers to how much a container can hold.

  • Relationship between Volume and Capacity: 1 mL = 1 cm³ and 1 L = 1000 cm³.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example 8 illustrates how to find the height of a cuboid given its volume and base area.

  • Example 9 demonstrates calculating how many smaller boxes fit into a larger storage area by dividing volumes.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • Volume measures space so wide, capacity counts what you can slide.

📖 Fascinating Stories

  • Imagine a jug that can hold 2 liters of water. The jug takes up space on the shelf (volume), and the amount of water it can hold (capacity) reminds us of how they work together.

🧠 Other Memory Gems

  • V = Volume, C = Capacity, M = Measure. Remember: Measure how much each can occupy!

🎯 Super Acronyms

VICs

  • Volume Indicates Capacity.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Volume

    Definition:

    The amount of space occupied by a three-dimensional object.

  • Term: Capacity

    Definition:

    The quantity that a container can hold, often measured in liters.

  • Term: Cubic Centimeter (cm³)

    Definition:

    A metric unit of volume equal to a cube measuring 1 cm on each side.

  • Term: Liter (L)

    Definition:

    A metric unit of capacity, equal to 1000 cm³.

  • Term: Meter Cubed (m³)

    Definition:

    A unit of volume in the metric system, equal to 1,000,000 cm³.