calculate ratio of the area to volume for unit cube, a unit sphere inscribed inside the cube and as right cylinder inscribed inside the cube
sphere diameter is the side of the cube, so radius is 1/2.
cylinder diameter and height are the side of the cube, so r = 1/2, h=1
cube: v=1 a=6, so 6:1
sphere: v=4/3 pi (1/2)^3 = pi/6
a = 4pi(1/2)^2 = pi
so, 6:1
cylinder: v = pi (1/2)^2 = pi/4
area= pi/2 + 2pi(1/2)(1) = 3pi/2
so, 6:1
whoda thunk it?
a cone is inscribed in a hemisphere which is inscribed inside a cylinder. show that the ratio of the volumes of cone to hemisphere to cylinder is 1:2:3
To calculate the ratio of the area to volume for different shapes, we need to find the formulas for the area and volume of each shape. Let's start with the unit cube.
1. Unit Cube:
A unit cube has all sides of length 1. The formula for the surface area of a cube is given by:
Surface area = 6 * (side length)^2
In this case, the side length is 1. Substituting the value in the formula, we get:
Surface area of the unit cube = 6 * (1)^2 = 6
The formula for the volume of a cube is given by:
Volume = (side length)^3
Substituting the value of side length, we get:
Volume of the unit cube = (1)^3 = 1
Therefore, the ratio of the area to volume for the unit cube is 6:1.
2. Unit Sphere Inscribed inside the Cube:
A unit sphere is a sphere with a radius of 1. The formula for the surface area of a sphere is given by:
Surface area = 4π * (radius)^2
In this case, the radius is 1. Substituting the value in the formula, we get:
Surface area of the unit sphere = 4π * (1)^2 = 4π
The formula for the volume of a sphere is given by:
Volume = (4/3)π * (radius)^3
Substituting the value of the radius, we get:
Volume of the unit sphere = (4/3)π * (1)^3 = (4/3)π
Therefore, the ratio of the area to volume for the unit sphere is 4π : (4/3)π, which simplifies to 3 : 1.
3. Right Cylinder Inscribed inside the Cube:
A right cylinder inscribed inside the cube will have a height and radius equal to half the length of a side of the cube.
The formula for the surface area of a cylinder is given by:
Surface area = 2π * (radius) * (height) + 2π * (radius)^2
Since the height and radius are equal in this case, let's call them 'r'. The surface area formula becomes:
Surface area of the right cylinder = 2π * r * r + 2π * r^2 = 2πr^2 + 2πr^2 = 4πr^2
The formula for the volume of a cylinder is given by:
Volume = π * (radius)^2 * (height)
In this case, the height is also 'r'. Substituting the values, we get:
Volume of the right cylinder = π * r^2 * r = πr^3
Therefore, the ratio of the area to volume for the right cylinder is 4πr^2 : πr^3, which simplifies to 4 : r.
Since the right cylinder is inscribed inside the cube, the radius 'r' will be half the length of a side of the cube, which is 1/2.
Substituting the value of 'r', we get:
Ratio of the area to volume for the right cylinder inscribed inside the cube = 4 : 1/2 = 8 : 1.
In summary, the ratios for the three shapes are:
- Unit cube: 6 : 1
- Unit sphere: 3 : 1
- Right cylinder inscribed inside the cube: 8 : 1