Each solid in the image has a height of 4 units. The area of each solid’s base is 8 square units. A cross-section has been created in each by dilating the base using the apex as a center with a scale factor of \(k=0.25\).

1. Calculate the area of each of the 2 cross-sections.
2. Suppose a new cross-section was created in each solid, both at the same height, using some scale factor \(k\). How would the areas of these 2 cross-sections compare? Explain your reasoning.

A solid can be constructed with 4 triangles and 1 rectangle. What is the name for this solid?

Find the volume of the solid produced by rotating this two-dimensional shape around the axis shown.

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This zigzag crystal vase has a height of 20 centimeters. The cross-sections parallel to the base are always rectangles that are 12 centimeters wide by 6 centimeters long.

1. If we assume the crystal itself has no thickness, what would be the volume of the vase?
2. The crystal is actually 1 centimeter thick on each of the sides and on the bottom. Approximately how much space is contained within the vase? Explain or show your reasoning.

A trapezoid has an area of 10 square units. What scale factor would be required to dilate the trapezoid to have an area of 90 square units?