The shortest path is along two of the surfaces of the prism. There are three possibilities, each of which are shown below with the relevant faces folded out. 

The top diagram shows a path along the right and the front. The path is equal in length to that of a diagonal of a rectangle of length and height 12 and 30, so the length of the path can be calculated using the Pythagorean Theorem. Setting : 

The bottom diagrams show a path along the top and the front and a path along the rear and the bottom. In both cases, the path is equal in length to that of a diagonal of a rectangle of length and height 20 and 22, so the length of the path can be calculated using the Pythagorean Theorem. Setting :

The shortest distance between Points A and B along the surface is therefore , or, if simplified using the Product of Radicals Rule:

.

Since the volume is the product of length, width, and height, and each of these three dimensions are integers, it is important to know the factors of the volume.  70 = (2)(5)(7).  This implies that each of these factors (and only these factors with the exception of 1) will show up in the three dimensions exactly once.  This creates precisely the following five possibilities:

2, 5, 7

SA = 2((2)(5)+(2)(7)+(5)(7)) = 118

1, 7, 10

SA = 2((1)(7)+(1)(10)+(7)(10)) = 174

1, 5, 14

SA = 2((1)(5)+(1)(14)+(5)(14)) = 178

1, 2, 35

SA = 2((1)(2)+(1)(35)+(2)(35)) = 214

1, 1, 70

SA = 2((1)(1)+(1)(70)+(1)(70)) = 282

Let's call the side lengths of the box l, w, and h. We are told that l, w, and h must all be integer lengths greater than one. We are also told that the volume of the box is 182 cubic units.

Since the volume of a rectangular box is the product of its side lengths, this means that lwh must equal 182. 

(l)(w)(h) = 182.

In order to determine possible values of l, w, and h, it would help us to figure out the factors of 182. We want to express 182 as a product of three integers each greater than 1. 

Let's factor 182. Because 182 is even, it is divisible by 2.

182 = 2(91).

91 is equal to the product of 7 and 13.

Thus, 182 = 2(7)(13).

This means that the lengths of the box must be 2, 7, and 13 units. 

In order to find the surface area, we can use the following formula:

surface area = 2lw + 2lh + 2hw.

surface area = 2(2)(7) + 2(2)(13) + 2(7)(13)

= 28 + 52 + 182

= 262 square units.

The answer is 262.

There are six faces to a right, rectangular prism. Based on our dimensions, we know that we must have a face that is 3 x 5, a face that is 5 x 20 and a face that is 3 x 20. To think this through, imagine that the front face is 3 x 5, the right side is 5 x 20, and the top is 3 x 20. Now, each of these sides has a matching side opposite (the left has the right, the top has the bottom, the front has the back).

Therefore, we know we have the following areas for the faces of our prism:

2 * 3 * 5 = 30

2 * 5 * 20 = 200

2 * 3 * 20 = 120

Add these to get the total surface area:

30 + 200 + 120 = 350

There are six faces to a right, rectangular prism. Based on our dimensions, we know that we must have a face that is 12.4 x 2.3, a face that is 2.3 x 33 and a face that is 33 x 12.4. To think this through, imagine that the front face is 12.4 x 2.3, the left side is 2.3 x 33, and the top is 33 x 12.4. Now, each of these sides has a matching side opposite (the left has the right, the top has the bottom, the front has the back).

Therefore, we know we have the following areas for the faces of our prism:

2 * 12.4 * 2.3 = 57.04

2 * 2.3 * 33 = 151.8

2 * 12.4 * 33 = 818.4

Add these to get the total surface area:

57.04 + 151.8 + 818.4 = 1027.24

Based on our prompt, we can say that the prism has dimensions that can be represented as:

Dim1: x

Dim2: 2 * Dim1 = 2x

Dim3: 2 * Dim2 = 2 * 2x = 4x

More directly stated, therefore, our dimensions are: x, 2x, and 4x. Therefore, the volume is x * 2x * 4x = 216, which simplifies to 8x³ = 216 or x³ = 27. Solving for x, we find x = 3. Therefore, our dimensions are:

x = 3

2x = 6

4x = 12

Or: 3 x 6 x 12

Now, to find the surface area, we must consider that this means that our prism has sides of the following dimensions: 3 x 6, 6 x 12, and 3 x 12. Since each side has a "matching" side opposite it, we know that we have the following values for the areas of the faces:

2 * 3 * 6 = 36

2 * 6 * 12 = 144

2 * 3 * 12 = 72

The total surface area therefore equals: 36 + 144 + 72 = 252 square units.

Converting squared units is not difficult, though you have to be careful not to make a simple mistake. It is tempting to think you can merely divide the initial value (30,096) by 12, as though you were converting from inches to feet.

Begin by thinking this through as follows. In the case of a single dimension, we know that:

1 ft = 12 in   or   1 in = (1/12) ft

Now, think the case of a square with dimensions 1 ft x 1 ft. This square has the following dimensions in inches: 12 in x 12 in. The area is therefore 12 * 12 = 144 in². This holds for all two-dimensional conversions. Therefore, the two dimensional conversion equation is:

1 ft² = 144 in²   or   1 in² = (1/144) ft²

Based on this, we can convert our value 30,096 in² thus: 30,096/144 = 209 ft².

Converting squared units is not difficult, though you have to be careful not to make a simple mistake. It is tempting to think you can merely multiply the initial value (24) by 36, as though you were converting from yards to inches.

Begin by thinking this through as follows. In the case of a single dimension, we know that:

1 yd = 36 in

Now, think the case of a square with dimensions 1 yd x 1 yd. This square has the following dimensions in inches: 36 in x 36 in. The area is therefore 36 * 36 = 1296 in². This holds for all two-dimensional conversions. Therefore, the two dimensional conversion equation is:

1 yd² = 1296 in²

Based on this, we can convert our value 24 yd² thus: 24 * 1296 = 31,104 in².

First we must calculate the surface area of the cube. We know that there are six surfaces and each surface has the same area:

Now we will determine the amount of paint needed

The surface area of the prism can be broken into three rectangular sides and two equilateral triangular bases.

The area of the sides is given by:  , so for all three sides we get .

The equilateral triangle is also an equiangular triangle by definition, so the base has congruent sides of 6 in and three angles of 60 degrees.  We use a special right traingle to figure out the height of the triangle: 30 - 60 - 90.  The height is the side opposite the 60 degree angle, so it becomes or 5.196. 

The area for a triangle is given by  and since we need two of them we get .

Therefore the total surface area is .