For 1 through 10, what is the area and volume of the given shape, if the length of one side of the base is 6, the height is 8, and the slant height is 10? (Not all shapes will require all three numbers.)

The shape is a right prism with:

1. an equilateral triangle as the base

To find the area and volume of a right prism with an equilateral triangle as the base, we can use the following formulas:

Area of the base = (sqrt(3) / 4) * (side length)^2
Lateral surface area = perimeter of the base * height
Total surface area = Lateral surface area + 2 * Area of the base
Volume = Area of the base * height

Given that the length of one side of the base is 6, we can substitute this value into the formulas.

1. Area of the base:
Area = (sqrt(3) / 4) * (side length)^2
= (sqrt(3) / 4) * (6)^2
= (sqrt(3) / 4) * 36
= 9sqrt(3)

2. Lateral surface area:
Since the base is an equilateral triangle, the perimeter is 3 * side length.
Perimeter of the base = 3 * 6
= 18

Lateral surface area = perimeter of the base * height
= 18 * 8
= 144

3. Total surface area:
Total surface area = Lateral surface area + 2 * Area of the base
= 144 + 2 * (9sqrt(3))
= 144 + 18sqrt(3)

4. Volume:
Volume = Area of the base * height
= 9sqrt(3) * 8
= 72sqrt(3)

Therefore, for the given right prism with an equilateral triangle as the base, the area of the base is 9sqrt(3), the lateral surface area is 144, the total surface area is 144 + 18sqrt(3), and the volume is 72sqrt(3).