The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area  square centimeters.

The area of an equilateral triangle is given by the formula

Set  and solve for : 

 centimeters.

The cross-section of the pool is the area of its surface, which is the product of its length and its width:

 square feet.

Since 80% of the pool is six feet deep, this portion of the pool holds 

 cubic feet of water.

Since the remainder of the pool - 20% - is four feet deep, this portion of the pool holds 

 cubic feet of water.

Add them together: the pool holds 

 cubic feet of water.

The tetrahedron has four faces, each of which is an equilateral triangle with sidelength . Since the total surface area is 444, each triangle has area one fourth of this, or 111. To find , set  in the formula for the area of an equilateral triangle:

We are looking for the height of the pyramid.

The base is an equilateral triangle with sidelength 4, so its area can be calculated as follows:

The height  of a pyramid can be calculated using the fomula

We set  and  and solve for :

The height of the pyramid is . The base is an equilateral triangle with sidelength 4, so its area can be calculated as follows:

The volume of a pyramid can be calculated using the fomula

The volume of a tetrahedron is found with the equation , where  represents the length of an edge of the tetrahedron.

Plug in 4 for the edge length and reduce as much as possible to find the answer:

The volume of the tetrahedron is .

The area of an equilateral triangle is given by the formula

Since there are four equilateral triangles that comprise the surface of the tetrahedron, the total surface area is 

Substitute :

 square inches.

The tetrahedron has four faces, each of which is an equilateral triangle with sidelength 7. Each face has area

The total surface area is four times this, or about .

Rounded, this is 84.9.

First we write out the equation we are given. H + (2L +2W) = 360.  H = 40 and L = 23

40 + (2(23) + 2W) = 360

40 + (46 + 2W) = 360

46 + 2W = 320

2W = 274

W = 137

Method 1:

Trial and error to find a combination of factors of 80 that differ by the same amount will eventually yield 2, 5, 8.  The average is 5.

Method 2:

Three terms of an arithmetic sequence can be written as x, x+d, and x+2d. Multiply these together using the distributive property to find the volume and the following equation results:

x³ + 3dx² + 2d²x - 80 = 0

Find an integer value of x that creates an integer solution for d.  Try x=1 and we see the equation 1 + 3d + 2d² - 80 = 0 or 2d² + 3d -79 = 0.  The determinant of this quadratic is 641, which is not a perfect square.  Therefore, d is not an integer when x=1.

Try x=2 and we see the equation 8 + 12d + 4d² - 80 = 0 or d² + 3d - 18 = 0.  This is easily factored to (d+6)(d-3)=0 so d=-6 or d=3.  Since a negative value of d will result in negative dimensions of the prism, d must equal 3.  Therefore, when substituting x=2 and d=3, the dimensions x, x+d, and x+2d become 2, 5, and 8.  The average is 5.