Question

The base radius and height of a solid metal cylinder are 8 centimetres and 15 centimetres. By melting it and recasting, how many cones of base radius 6 centimetres and slant height 10 centimetres can be made ?

Hint:

In problems involving "melting and recasting", the fundamental principle is the conservation of volume. Always remember to calculate the perpendicular height of a cone if the slant height is given, as volume formulas use perpendicular height. Recognize Pythagorean triples like 3-4-5 (and its multiples like 6-8-10) to speed up calculations.

Answer 1

We are melting a cylinder and recasting it into smaller cones. The volume of the material remains constant. We need to find the number of cones that can be made by dividing the volume of the cylinder by the volume of a single cone.

- Volume of a cylinder: Vcyl = π r² h
- Volume of a cone: Vcₒₙₑ = (1)/(3) π r² h
- For a cone, the relationship between slant height (l), radius (r), and perpendicular height (h) is l² = r² + h².

1. Calculate the volume of the cylinder.
Given: Radius of cylinder, Rcyl = 8 cm, and Height of cylinder, Hcyl = 15 cm.
Vcyl = π (Rcyl)² Hcyl = π × (8)² × 15 = π × 64 × 15 = 960π cm³ 2. Calculate the volume of one cone.
Given: Radius of cone, rcₒₙₑ = 6 cm, and Slant height of cone, lcₒₙₑ = 10 cm.
First, we must find the perpendicular height (hcₒₙₑ) of the cone.
lcₒₙₑ² = rcₒₙₑ² + hcₒₙₑ² 10² = 6² + hcₒₙₑ² 100 = 36 + hcₒₙₑ² hcₒₙₑ² = 100 - 36 = 64 hcₒₙₑ = √(64) = 8 cm Now, we can calculate the volume of the cone:
Vcₒₙₑ = (1)/(3) π (rcₒₙₑ)² hcₒₙₑ = (1)/(3) π × (6)² × 8 = (1)/(3) π × 36 × 8 = 12π × 8 = 96π cm³ 3. Find the number of cones.
Number of cones = Volume of CylinderVolume of one Cone = VcylVcₒₙₑ Number of cones = (960π)/(96π) = (960)/(96) = 10 Exactly 10 cones can be made from the melted cylinder.