Question

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.

Answer 1

The cylindrical box is about 1.5 times larger in volume compared to the sphere. The ratio of the sphere's surface area to the cylinder's surface area is 2/3. The largest possible cone that could fit inside the same box has a volume of 2/3 times the volume of the sphere.

Step-by-step explanation:

To find the volume of the cylindrical box, we need to know the radius of the sphere. Let's assume the radius of the sphere is r. The volume of the sphere is given by the formula:

Vsphere = (4/3)×πr3

The volume of the cylindrical box is given by the formula:

Vcylinder = πr2h

Since we want the cylindrical box to fit the sphere perfectly, we can assume that the height of the cylinder will be equal to the diameter of the sphere, which is 2r. Therefore, the volume of the cylindrical box can be written as:

Vcylinder = πr2(2r) = 2πr3

To compare the volumes, we can calculate the ratio of the box's volume to the sphere's volume:

Ratio of volumes: (Vcylinder)/(Vsphere) = (2πr3)/((4/3)×πr3) = 3/2

To compare the surface areas, we can use the formula:

Surface area of the sphere: Asphere = 4πr2

Surface area of the cylinder: Acylinder = 2πr(2r) + 2πr2 = 4πr2 + 2πr2 = 6πr2

The ratio of the sphere's surface area to the cylinder's surface area is:

Ratio of surface areas: (Asphere)/(Acylinder) = (4πr2)/(6πr2) = 2/3

To find the largest possible cone that could fit inside the same box, we need to consider the height of the cone. Let's assume the height of the cone is hcone. The volume of the cone is given by the formula:

Vcone = (1/3)πr2hcone

Since we want the cone to fit inside the same box, the height of the cone should be less than or equal to the height of the cylinder, which is 2r. Therefore, the largest possible cone that could fit inside the box would have a height of 2r. The volume of this cone would be:

Vcone = (1/3)πr2(2r) = (2/3)πr3

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