Solved Problems: Applying Volume to Everyday Scenarios (Grade 7)

📚 Understanding Volume: A Practical Guide

Volume is the amount of space a three-dimensional object occupies. It's a fundamental concept in geometry and has countless applications in our daily routines.

📜 A Little History of Volume

The concept of volume has been around for centuries, dating back to ancient civilizations like the Egyptians and Greeks. They needed to calculate volumes for construction projects, irrigation, and even for measuring grain. Archimedes, a famous Greek mathematician, made significant contributions to understanding volume, particularly with his work on buoyancy and displacement.

• 🏺 Archimedes famously determined the purity of a gold crown by measuring its volume and comparing it to the volume of an equal weight of pure gold.
• 📐 Early Egyptians used formulas to calculate the volume of pyramids and other structures.

📌 Key Principles of Volume

The basic principle of volume calculation involves multiplying the area of the base of an object by its height. For different shapes, we use specific formulas:

• 🧊Cube: Volume = $side \times side \times side = s^3$
• 📦Rectangular Prism: Volume = $length \times width \times height = lwh$
• 🥤Cylinder: Volume = $\pi \times radius^2 \times height = \pi r^2 h$
• 🍦Cone: Volume = $\frac{1}{3} \times \pi \times radius^2 \times height = \frac{1}{3} \pi r^2 h$
• ⚽Sphere: Volume = $\frac{4}{3} \times \pi \times radius^3 = \frac{4}{3} \pi r^3$

🏠 Real-World Examples: Volume in Action

Let's see how volume is useful in everyday situations:

• 🐠 Fish Tank: You need to determine how much water is required to fill a fish tank that is 30 cm long, 20 cm wide, and 20 cm high. Volume = $30 \times 20 \times 20 = 12000 \text{ cm}^3$. Therefore, you need 12 liters of water (since 1000 cm³ = 1 liter). 🌊
• 🍿 Popcorn Bowl: A cylindrical popcorn bowl has a radius of 7 cm and a height of 15 cm. How much popcorn can it hold? Volume = $\pi \times 7^2 \times 15 \approx 2310 \text{ cm}^3$. 🍿
• 📦 Shipping Box: A shipping box is 50 cm long, 30 cm wide, and 40 cm high. Its volume is $50 \times 30 \times 40 = 60000 \text{ cm}^3$ or 60 liters. This tells you how much space you have for packing items. 🚚

📝 Practice Quiz

Test your understanding with these volume problems:

1. 🧱 A rectangular brick is 20 cm long, 10 cm wide, and 8 cm high. What is its volume?
2. 🎂 A cylindrical cake tin has a radius of 10 cm and a height of 5 cm. What is its volume?
3. 🎁 A cube-shaped gift box has sides of 12 cm. What is its volume?

Answers:

1. $20 \times 10 \times 8 = 1600 \text{ cm}^3$
2. $\pi \times 10^2 \times 5 \approx 1570 \text{ cm}^3$
3. $12 \times 12 \times 12 = 1728 \text{ cm}^3$

💡 Conclusion

Understanding volume is incredibly practical. It helps us with everyday tasks and provides a foundation for more advanced mathematical and scientific concepts. Keep practicing, and you'll master it in no time! 🎉