Suppose a sphere, cylinder and cone all share the same radious and the cylinder and cone also share the same height which is true about the retationship

Answer 1

Let’s analyze the relationships between the volumes and surface areas of a sphere, a cylinder, and a cone, all sharing the same radius \( r \) and with the cylinder and cone having the same height \( h \).

Formulas:

1. Sphere:

   • Volume: \( V_s = \frac{4}{3} \pi r^3 \)
   • Surface Area: \( A_s = 4 \pi r^2 \)

2. Cylinder:

   • Volume: \( V_c = \pi r^2 h \)
   • Surface Area: \( A_c = 2 \pi r (r + h) \)

3. Cone:

   • Volume: \( V_{cone} = \frac{1}{3} \pi r^2 h \)
   • Surface Area: \( A_{cone} = \pi r (r + l) \) where \( l = \sqrt{r^2 + h^2} \) (the slant height).

Relationships:

1. Volume:

   • Sphere vs. Cylinder: \( V_s = \frac{4}{3} \pi r^3 \) vs. \( V_c = \pi r^2 h \).
     • If \( h = 2r \), then \( V_c = 2 \pi r^3 \) and we see that the volume of the sphere will be \( \frac{4}{3} \pi r^3 \) which is greater than \( V_c \).
   • Cylinder vs. Cone: \( V_c = \pi r^2 h \) vs. \( V_{cone} = \frac{1}{3} \pi r^2 h \).
     • Thus, \( V_c = 3 V_{cone} \). The volume of the cylinder is three times that of the cone.

2. Surface Area:

   • For the Sphere and the Cylinder:
     • \( A_s = 4 \pi r^2 \)
     • \( A_c = 2 \pi r (r + h) = 2 \pi r^2 + 2 \pi r h \) depending on the specific height \( h \).
   • For the Cone, using the slant height \( l \):
     • \( A_{cone} = \pi r (r + l) \).
     • If \( h \) is related to \( l \), it also could be equal to or greater than the sphere's surface area when \( h \) increases significantly.

Summary:

1. The volume of the sphere is generally greater than that of the cylinder when \( h < 2r \) but can be equal if conditions adjust.
2. The volume of the cylinder is three times that of the cone when both share the same height \( h \).
3. The surface area relationships depend on \( h \), with the sphere often providing a constant surface area that can be exceeded by the increasing surface areas of the cone or cylinder based on height.

This summary outlines key comparisons and relationships regarding volume and surface area among the three shapes under the described conditions. Further, specific numerical examples can clarify particular instances of relationships based on varied heights and radii.