Finding the Volume of a Frustum

Definition and Concept

• A frustum is a shape formed by chopping off the top of a cone.
• To find the volume of a frustum, calculate the volume of the larger cone and subtract the volume of the smaller cone.

Formula for Cone Volume

• Volume of a cone: ( \frac{1}{3} \pi r^2 h )
  • Use uppercase ( R ) and ( H ) for the large cone.
  • Use lowercase ( r ) and ( h ) for the small cone.

Steps to Calculate Frustum Volume

1. Identify Cone Dimensions

   • Large Cone
     • Height ( H = 50 ) cm
     • Radius ( R = 10 ) cm
   • Small Cone
     • Height ( h = 30 - 50 = 20 ) cm
     • Radius needs to be calculated.

2. Calculate Scale Factor

   • Scale factor = ( \frac{\text{Height of large cone}}{\text{Height of small cone}} = \frac{50}{20} = 2.5 )
   • Small cone is 2.5 times smaller.

3. Calculate Radius of Small Cone

   • Radius ( r = \frac{\text{Radius of large cone}}{\text{Scale factor}} = \frac{10}{2.5} = 4 ) cm

4. Plug Values into Formula

   • Volume of large cone: ( \frac{1}{3} \pi (10)^2 (50) )
   • Volume of small cone: ( \frac{1}{3} \pi (4)^2 (20) )
   • Frustum Volume = Volume of large cone - Volume of small cone
     • Simplify: ( \frac{5000}{3} \pi - \frac{320}{3} \pi = 1560 \pi \text{ cm}^3 )
     • Approximate: 4900 cm³ to 3 significant figures

Conclusion

• Practicing similar questions can help solidify understanding.
• Additional resources available on the platform.

This is a concise guide to finding the volume of a frustum by using the cone volume formula and applying the concept of similar shapes to calculate the unknown dimensions. Perfect for quick revision or homework help!