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? shape of base side length height square cm cm regular hexagon cm cm

Pyramid is a three dimensional Structure ,having square , rectangle , hexagonal base , the faces for a square pyramid and hexagonal pyramid are triangle meeting at one point .

It is given in the question that

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

Let us assume the side length to be x

and therefore the height of the toy is 2x

The surface area of the toy is given = 250 sq.cm

First let us consider for a square base

The surface area of a square pyramid is given by

[tex]\rm A = a^2 + 2 a\sqrt {\dfrac{a^2}{4}+h^2}[/tex]

Here a is the side length which x for us and h = 2x

[tex]\rm 250 = x^2 + 2 x\sqrt {\dfrac{x^2}{4}+(2x)^2}\\\\ 250 = x^2 + 2 x\sqrt {\dfrac{x^2}{4}+4x^2}\\\\\\ 250 = x^2 + 2 x\sqrt {\dfrac{17 x^2}{4}}\\\\\\250 = x^2 + x^2 \sqrt{17}\\\\\\250 = 5.132 x^2\\\\\\48.71 = x^2\\\\\\x = 6.98 cm .[/tex]

Therefore the length of the side of the square and the height will be

6.98 cm and 13.96 cm.

For a hexagonal base

The total surface area is given by

[tex]A =6ah + 3 \sqrt{3}a^2\\\\\\A = 6 \times x \times 2x + 3\sqrt{3} (x)^2\\\\250= 12x^2 + 3\sqrt{3} x^2\\\\\\250 = 17.196 x^2\\\\\\14.54 = x^2\\\\x = 3.81 cm\\\\[/tex]

Therefore the side of the hexagonal pyramid will be 3.81 cm and the height of the toy will be 7.62 cm.

To know more about Pyramid

https://brainly.com/question/13057463

#SPJ1