Find the Volume of a Cylinder Inscribed in a Sphere

Finding the Volume: Cylinder Inside a Sphere

Let's break down how to find the volume of a right circular cylinder inscribed in a sphere.

Problem: A right circular cylinder of radius 'r' is inscribed in a sphere of radius '4r'. Find a formula for 'V', the volume of the cylinder, in terms of 'r'.

Understanding the Setup

• Inscribed Cylinder: This means the cylinder fits perfectly inside the sphere, touching the sphere's surface along its circular top and bottom.* Key Relationship: The diameter of the cylinder is equal to the diameter of the sphere. This means the cylinder's height is twice the sphere's radius minus the cylinder's radius.

Solution

1. Cylinder's Height (h): * Sphere's Diameter = 2 * Sphere's Radius = 2 * 4r = 8r * Cylinder's Height (h) = Sphere's Diameter - 2 * Cylinder's Radius = 8r - 2r = 6r

2. Cylinder's Volume (V): * Formula: V = πr²h * Substitute: V = πr²(6r) = 6πr³

Therefore, the correct formula for the volume of the cylinder in terms of 'r' is V = 6πr³.

Important Note: The options provided (A, B, C, and D) in the original question are all incorrect.