Multiple choice

Consider a pyramid with a square base side of $6$ inches and with a height of $12$ inches, as shown below. If we cut off the top of the pyramid parallel to the base $3$ inches from the tip, what is the volume of the remaining solid?

  1. $141.75$
  2. $140$
  3. $135.48$
  4. $144$
  5. $130$
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A Correct answer
Explanation

The original pyramid has height 12 and base side 6. The cut is made 3 inches from the tip, creating a smaller pyramid with height 3. By similar triangles, the base side of the smaller pyramid is (3/12) * 6 = 1.5. Volume = (1/3) * base_area * height. Original volume = (1/3) * 36 * 12 = 144. Smaller volume = (1/3) * (1.5^2) * 3 = 2.25. Remaining volume = 144 - 2.25 = 141.75.

AI explanation

The volume of a pyramid is found using the formula (1/3) * base_area * height. The original pyramid has a volume of (1/3) * (6 * 6) * 12, which equals 144 cubic inches. The removed top part is a smaller, similar pyramid with a height of 3 inches, meaning its linear dimensions are 3/12 = 1/4 of the original, so its volume is (1/4)^3 = 1/64 of the original volume. The volume of the removed top is 144/64 = 2.25, making the volume of the remaining solid 144 - 2.25. The final volume of the remaining solid is 141.75.

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