Question

# A pyramid with a square base is cut by a plane that is parallel to its base and is 2 units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid

Answer

The altitude of the original pyramid is 6 units.

- Q: What is the formula for the surface area of a pyramid? A: The formula is S = (1/2)Pl + B, where P is the perimeter of the base, l is the slant height, B is the area of the base.
- Q: What is the formula for the volume of a pyramid? A: The formula is V = (1/3)Bh, where B is the area of the base and h is the height.
- Q: What is the formula for the slant height of a pyramid? A: The formula is l = sqrt(h^2 + (1/4)s^2), where h is the height of the pyramid and s is the length of one side of the base.
- Q: What is the relationship between the surface area of the smaller pyramid and the original pyramid? A: The surface area of the smaller pyramid is half the surface area of the original pyramid.
- Q: What is the formula for the altitude of the original pyramid? A: The formula is h = sqrt(l^2 - (1/4)s^2), where l is the slant height and s is the length of one side of the base.
- Q: What are the values we know for the original pyramid? A: We know that the smaller pyramid has a surface area of half the original pyramid and that the smaller pyramid has a height of 2 units. We also know that the base of the original pyramid is a square.
- Q: How do we use this information to find the altitude of the original pyramid? A: We can use the formula for the surface area of a pyramid and the fact that the surface area of the smaller pyramid is half the original pyramid to find the length of one side of the base. We can then use the formula for the slant height of a pyramid to find the height of the original pyramid. Finally, we can use the formula for the altitude of a pyramid to find the altitude of the original pyramid.