Mrs. Kertye wants to pack a cone-shaped toy inside the same box. What is the largest possible cone that could fit inside? Give its volume.

Question

A cylinder, rendered in a light translucent blue, is shown containing a sphere, also in a light translucent blue. 

The cylinder has thick black outlines, and the sphere is defined by two dashed black lines indicating the top and bottom circular sections of the sphere touching the cylinder. These sections are slightly smaller in diameter than the cylinder's opening.

Inside the cylinder, a sphere of the same translucent blue is visible, with its upper and lower sections inside the cylinder's boundaries. 

A thin, solid black line represents the radius (r) of the sphere, extending from the center of the sphere to the edge of the sphere's circumference at the level within the cylinder, marked with a small black circle to denote the sphere's center.

Answer 1

To determine the largest possible cone that can fit inside a cylinder, we first need to understand the dimensions of both the cylinder and the sphere contained within it.

1. Dimensions of the Cylinder:

   • The cylinder has a height \( h \) and a radius \( R \).
   • Since the sphere fits snugly within the cylinder, the diameter of the sphere is equal to the diameter of the cylinder, so \( R = r \) where \( r \) is the radius of the sphere.

2. Dimensions of the Cone:

   • For the largest cone that can fit inside the cylinder, the cone will have its base on the bottom of the cylinder and its apex at the top (or at the highest point inside the cylinder if the cone is fully contained within).
   • The radius of the cone’s base will also be \( R \), and its height will be \( h \) (the same as that of the cylinder).

3. Volume of the Cone: The volume \( V \) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.

Substituting \( R \) for \( r \) (the radius of the cone’s base) and \( h \) for the height: \[ V = \frac{1}{3} \pi R^2 h \]

4. Final Result: To find the specific volume, you simply need the values of \( R \) (the radius of the cylinder) and \( h \) (the height of the cylinder). With these dimensions provided, you can calculate the volume of the largest possible cone that fits inside the cylinder.

Let’s summarize:

• The volume of the largest cone that can fit inside the cylinder is given by \( V = \frac{1}{3} \pi R^2 h \).
• Substitute the specific measurements for \( R \) and \( h \) to find the exact volume.

If actual measurements (like \( R \) and \( h \)) are included, you can then substitute those numbers into the formula to obtain a numeric volume.