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A toy is being constructed in the shape of a pyramid. The maximum amount of material to cover the sides and bottom of the pyramid is 250 square centimeters. The height of the toy is double the side length. What are the maximum dimensions to the nearest square centimeter for a square base and for a hexagonal base?.

Answer :

The maximum dimensions (side length and height) of the pyramid for a given square base is equal to 6.99cm and 13.98cm respectively.

As given in the question,

Maximum amount of material used to cover sides of pyramid and its bottom = 250cm²

Surface area of the ( sides + bottom ) = 250cm²

Base is square :

Let 's' represent the the side length

Then height 'h' of the toy = 2 ( side length)

                                           = 2s

Surface area of Pyramid = s² + 2s√(s²/4)  + h²

⇒ 250 = s² + 2s√(s²/4)  + (2s)²

⇒250 = s² + 2s√(s² + 16s²)/4

⇒250 = s² + 2s√17s²/4

⇒250 = s² + 2s(s/2)√17

⇒250 = s²+ s²√17

⇒250 = s² ( 1+ √17)

⇒250 = s²( 1 + 4.123 )

⇒ 250 = 5.123s²

⇒s² = 250/5.123

⇒ s= 6.99cm

Height 'h' = 2(6.99)

                = 13.98cm

Therefore, the side-length and height of the pyramid with square base is equal to 6.99cm and 13.98cm respectively.

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