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.
About how much larger is the cylindrical box’s volume compared to the ball’s volume?
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?
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.
To find out how much larger the cylindrical box's volume is compared to the ball's volume, we need to know the formula for the volume of a sphere and the volume of a cylinder.
The volume of a sphere is given by:
V_sphere = (4/3) * π * r^3
The volume of a cylinder is given by:
V_cylinder = π * r^2 * h
To compare the volumes, we'll need to know the radius of the ball, which is not given in the problem. Therefore, we cannot determine the exact difference in volumes.
Regarding their surface areas, we need to know the formulas for the surface area of a sphere and the surface area of a cylinder.
The surface area of a sphere is given by:
A_sphere = 4 * π * r^2
The surface area of a cylinder is given by:
A_cylinder = 2 * π * r * (r + h)
Again, without knowing the radius of the ball or the height of the cylinder, we cannot determine the exact ratio of their surface areas.
As for packing a cone-shaped toy inside the same box, we'll need to know the radius and height of the cone. With that information, we can calculate the largest possible cone that could fit inside the box.
The volume of a cone is given by:
V_cone = (1/3) * π * r^2 * h
Without specific values for the radius and height of the cone, we also cannot determine its volume.