A sphere is contained inside of a cube, tangent to all six faces. Find the surface area of the cube as a function of the radius of the sphere.


The answer in the back of the book read:A=24r^2...How did they get this answer? Thanks for helping!

Say the radius of the sphere is r. The sphere touches both sides of the cube, so the distance between two faces of the cube is 2r. But that also equals one edge, so one side of the cube has area 2r x 2r = 4r². But a cube has six such sides...

Well, it seems like this problem is trying to test your geometry skills! To find the surface area of the cube, we need to consider that the sphere is tangent to all six faces of the cube.

Each face of the cube is a square, and since the sphere is tangent to all the faces, the diagonal of each face will be equal to the diameter of the sphere (which is twice the radius).

Now, let's focus on one face of the cube. Since the diagonal of the face is equal to the diameter of the sphere, which is 2r, we can use the Pythagorean theorem to find the side length of the cube.

If we let "s" represent the side length of the cube, then using the Pythagorean theorem, we have:

s^2 + s^2 = (2r)^2
2s^2 = 4r^2
s^2 = 2r^2
s = √(2r^2)

Since there are 6 faces in total, the total surface area of the cube can be found by multiplying the area of one face by 6.

So, the surface area of the cube would be:

A = 6s^2
A = 6(√(2r^2))^2
A = 6(2r^2)
A = 12r^2

So, the answer given in the book appears to be incorrect. The correct surface area of the cube as a function of the radius of the sphere should be A = 12r^2.

To find the surface area of the cube as a function of the radius of the sphere, we can follow these steps:

1. Let's start by assuming that the side length of the cube is 's'.

2. Since the sphere is tangent to all six faces of the cube, the diameter of the sphere is equal to the side length of the cube. Therefore, the diameter of the sphere is also 's'.

3. The radius of the sphere is half the diameter, so we have: r = s/2.

4. Now, let's calculate the surface area of the cube. The surface area of a cube is given by the formula: A = 6s^2.

5. Substituting the value of 's' from step 3, we get: A = 6( (2r)^2).

6. Simplifying further, we have: A = 6(4r^2) = 24r^2.

Therefore, the surface area of the cube is given by the function A = 24r^2.

To find the surface area of the cube, we need to find the total area of all six faces. Let's break it down step by step:

1. Start by visualizing the cube with the sphere inside it. Since the sphere is tangent to all six faces of the cube, it touches each face at exactly one point.

2. Each face of the cube is a square, and since the sphere touches each face at exactly one point, the side length of each face is equal to the diameter of the sphere.

3. Let's denote the radius of the sphere as "r". Since the diameter is twice the radius, the side length of each face of the cube is 2r.

4. Now, calculate the area of one face of the cube. The formula for finding the area of a square is side length squared. So, the area of one face is (2r)^2 = 4r^2.

5. Since all six faces of the cube have the same area, we need to multiply the area of one face by 6 to find the total surface area of the cube.

6. Multiply 4r^2 by 6: 4r^2 * 6 = 24r^2.

Therefore, the surface area of the cube, as a function of the radius of the sphere, is given by A = 24r^2.