To begin, we need to solve for the dimensions of the cone.

The basic form for the volume of a cone is: V = (1/3)πr²h

Using our data, we know that h = 2r because the height of the cone matches its diameter (based on the prompt).

486π  = (1/3)πr² * 2r = (2/3)πr³

Multiply both sides by (3/2π): 729 = r³

Take the cube root of both sides: r = 9

Note that this is in feet. The answers are in square inches. Therefore, convert your units to inches: 9 * 12 = 108, then add 3 inches to this: 108 + 3 = 111 inches.

The surface area of the cube is defined by: A = 6 * s², or for our data, A = 6 * 111² = 73,926 in²

The volume of a cube is given by the formula V = , where V is the volume, and s is the length of each side. We can set V to 216 and then solve for s.

In order to find s, we can find the cube root of both sides of the equaton. Finding the cube root of a number is the same as raising that number to the one-third power.

This means the length of the side of the cube is 6. We can use this information to find the surface area of the cube, which is equal to . The formula for surface area comes from the fact that each face of the cube has an area of , and there are 6 faces in a cube.

surface area =

The surface area of the square is 216 square units.

The answer is 216.

The area of one side of the cube is:

A cube has 6 sides, so the total surface area of the cube is

We have a cube with a surface area of 24, which means each side has an area of 4. Therefore, the length of each side is 2. If we double the height, the volume becomes .

The equation for surface area of the original cube is 6s². If the sides all double in length the new equation is 6(2s)² or 6 * 4s². This makes the new surface area 4x that of the old. 4x10 = 40m²

The area of a cube is equal to the measure of one edge cubed. If we take the cube root of , we get , so the edges of this cube measure ; therefore, one face of the cube has an area of  square inches, because .

There are  sides to a cube, so

General formula for the diagonal of a cube if each side of the cube = s

Use Pythagorean Theorem to get the diagonal across the base:

s² + s² = h²

And again use Pythagorean Theorem to get cube’s diagonal, then solve for d:

h² + s² = d²

s² + s² + s² = d²

3 * s² = d²

d = √ (3 * s²) = s √3

So, if s = 3 then the answer is 3√3

Since the diagonal of the cube is a line segment that goes through the center of the cube (and also the circumscribed sphere), it is clear that the diagonal of the cube is also the diameter of the sphere.  Since the radius = 1, the diameter = 2.

The first thing necessary is to determine the dimensions of the cube.  This can be done using the volume formula for cubes: V = s³, where s is the length of the cube. For our data, this is:

s³ = 512, or (taking the cube root of both sides), s = 8.

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (8,8,8).  The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):

d = √( (x₁ – x₂)² + (y₁ – y₂)²  + (z₁ – z₂)²) 

Or for our simpler case:

d = √( (x)² + (y)²  + (z)²) = √( (s)² + (s)²  + (s)²) = √( (8)² + (8)²  + (8)²) = √( 64 + 64 + 64) = √(64 * 3) = 8√(3)

The first thing necessary is to determine the dimensions of the cube.  This can be done using the volume formula for cubes: V = s³, where s is the length of the cube. For our data, this is:

s³ = 1728, or (taking the cube root of both sides), s = 12.

The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (12,12,12).  The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean theorem):

d = √( (x₁ – x₂)² + (y₁ – y₂)²  + (z₁ – z₂)²) 

Or, for our simpler case:

d = √( (x)² + (y)²  + (z)²) = √( (s)² + (s)²  + (s)²) = √( (12)² + (12)²  + (12)²) = √( 144 + 144 + 144) = √(3 * 144) = 12√(3) = 12√(3)