ratio of volume and surface area of a cylinder inscribed in a sphere?

Question

A right cylinder whose height and diameter are equal is inscribed in a sphere with radius 'r'. Determine the ratio of the volumes of the two solids and the ratio of their surface areas.

Anybody think they can help me out? The answer shouldn't have any variables. Please reply soon!

Answer 1

Volume of a sphere is 4/3 pi * radius^3.
Volume of a cylinder is pi * radius^2 * height.

Cylinder:
Height = Diameter -> Height = 2*Radius or 1/2 Height = Radius
Thus, the volume for this cylinder can be written as:
pi * radius^2 * 2*radius or 2 pi * radius ^3.

For the ratio of volume of the sphere to the volume of the cylinder, it is (4/3 pi * radius^3) / (2 pi * radius ^3), simplified to 2/3.

Surface Area of a Sphere: 4 pi * radius^2
Surface Area of a Cylinder: (2 pi * radius^2) + (2 pi * radius * height)

Using the previously stated information that the height = 2 radius, the "SA" of the cylinder can be written as (2 pi * radius^2) + (2 pi * radius * 2 radius) or (2 pi * radius^2) + (4 pi * radius^2), which simplifies to (6 pi * radius^2).

Thus, the ratio of the "SA" of a sphere to the "SA" of a cylinder is (4 pi * radius^2) / (6 pi * radius^2), which also simplifies to 2/3.

Answer 2

Let
R = radius sphere
r = radius cylinder
h = height cylinder
V = volume sphere
C = volume cylinder

V = 4πR³/3

The height of the cylinder equals the diameter or twice the radius.

h = 2r

C = πr²h = πr²(2r) = 2πr³

When you inscribe the cylinder into the sphere you have a right triangle 45 90 45.

r² + r² = R²
2r² = R²
r² = R²/2
r = R/√2

C = 2πr³ = 2π(R/√2)³ = πR³/√2

The ratio of the volume of the cylinder to the volume of the sphere is:

C / V = (πR³/√2) / (4πR³/3) = 3/(4√2) ≈ 0.53033
___________________

Let
S = surface area sphere
A = surface area cylinder

S = 4πR²

A = 2πr² + 2πrh = 2πr² + 2πr(2r) = 2πr² + 4πr² = 6πr²
A = 6πr² = 6π(R/√2)² = 3πR²

The ratio of the area of the cylinder to the area of the sphere is:

A / S = 3πR² / (4πR²) = 3/4
____________

The answerer above me inscribed the sphere in the cylinder instead of the cylinder in the sphere.

Answer 3

Im undesirable at math mn it facilitates me if i positioned ex questions n positioned each and all of the stairs on a thank you to slove that variety of problum and if u dont have all waiting consistent with probability some voc if thats on there