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A pyramid has a square base. The side of square is 12 cm and height of pyramid is 21 cm. The pyramid is cut into 3 parts by 2 cuts parallel to its base. The cuts are at height of 7 cm and 14 cm respectively from the base. What is the difference (in cm3) in the volume of top most and bottom most part?

A pyramid has a square base. The side of square is 12 cm and height of pyramid is 21 cm. The pyramid is cut into 3 parts by 2 cuts parallel to its base. The cuts are at height of 7 cm and 14 cm respectively from the base. What is the difference (in cm3) in the volume of top most and bottom most part? Correct Answer <span style="display: none;"> </span>672<span style="display: none;"> </span>

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Using similarity of the triangle

⇒ ΔABP ≅ ΔADR

The ratio of the sides of two similar triangles is equal to the ratio of the corresponding sides

⇒ AB/AD = BP/DR

⇒ 7/21 = BP/6

⇒ BP = 2

Side of square EFGH is 4 cm

Similarly, ΔABP ≅ ΔACQ

⇒ AB/AC = BP/CQ

⇒ 7/14 = 2/CQ

⇒ CQ = 4

Side of square IJKL is 8 cm

⇒ Volume of top most part of pyramid (V1) = (1/3) × h' × (area of base EFGH)
⇒ Volume of top most part of pyramid (V1) = (1/3) × 7 × 4 × 4

⇒ Volume of top most part of pyramid (V1) = (7/3) × 16

⇒ Area of square IJKL (A1) = 8 × 8

⇒ A1 = 64 cm

⇒ Area of square STUV (A2) = 12 × 12

⇒ A2 = 144 cm

 ⇒ Volume of bottom part of pyramid (V) = (1/3) × h × (A1 + √(A1 × A2) + A2)

⇒ Volume of bottom part of pyramid (V) = (1/3) × 7 × (64 + √(64 × 144) + 144)

⇒ Volume of bottom part of pyramid (V) = (7/3) × 304

The difference in the volume of top most and bottom most part = V – V1

⇒ The difference in the volume of top most and bottom most part = (7/3) × 304 – (7/3) × 16

⇒ The difference in the volume of top most and bottom most part = (7/3) × (304 – 16)

∴ The difference in the volume of top most and bottom most part = 672 cm3 

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