Mrs. Kertye bought a spherical ball for her nephew, but she does not know the radius.
She is in search of a cylindrical box that will fit the ball perfectly, that is, it will have the same radius plus a little more so that the ball will fit.
1. About how much larger is the cylindrical box's volume compared to the ball's volume?
2. How do their surface areas compare to each other? What is the ratio of the sphere's surface area to the cylinder's surface area?
3. 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.
1. The volume of a sphere is given by the formula V = (4/3) * π * r^3, where r is the radius of the sphere. The volume of a cylindrical box is given by the formula V = π * r^2 * h, where r is the radius of the base and h is the height of the cylinder.
Since Mrs. Kertye does not know the radius of the ball, it is not possible to determine the exact ratio of the cylindrical box's volume to the ball's volume. However, we can assume that the cylindrical box's radius is slightly larger than the ball's radius to ensure a perfect fit. Thus, the cylindrical box's volume would be slightly larger than the ball's volume.
2. The surface area of a sphere is given by the formula A = 4 * π * r^2, where r is the radius of the sphere. The surface area of a cylinder is given by the formula A = 2 * π * r * (r + h), where r is the radius of the base and h is the height of the cylinder.
Similarly, since Mrs. Kertye does not know the radius of the ball, it is not possible to determine the exact ratio of the sphere's surface area to the cylinder's surface area. However, we can assume that the sphere's surface area would be smaller than the cylinder's surface area if the cylindrical box is larger than the ball.
3. To determine the largest possible cone that could fit inside the cylindrical box, we need to consider the dimensions of the box and the constraints of the cone. The cone's base must be able to fit inside the cylindrical box's base, and the height of the cone must be less than or equal to the height of the cylindrical box.
Assuming the height of the cylindrical box is greater than the height of the cone, we can calculate the volume of the cone using the formula V = (1/3) * π * r^2 * h, where r is the radius of the cone's base and h is the height of the cone. The largest possible cone that could fit inside the cylindrical box would have a radius equal to the radius of the cylindrical box's base and a height equal to the height of the cylindrical box.