To determine the maximum dimensions of the toy pyramid with a material limit of 250 square centimeters and a height that is double the side length, let's consider both a square base and a hexagonal base.
### 1. Square Base
For a pyramid with a square base:
- Let the side length of the square base be [tex]\( s \)[/tex] cm.
- The height of the pyramid is [tex]\( h \)[/tex] cm.
- Given that the height [tex]\( h \)[/tex] is double the side length [tex]\( s \)[/tex], we have [tex]\( h = 2s \)[/tex].
We'll solve for [tex]\( s \)[/tex]:
#### Solution:
- Side Length for Square base: [tex]\( 7 \)[/tex] cm
- Height for Square base: [tex]\( 14 \)[/tex] cm
### 2. Hexagonal Base
For a pyramid with a hexagonal base:
- Let the side length of the hexagonal base be [tex]\( s \)[/tex] cm.
- The height of the pyramid is [tex]\( h \)[/tex] cm.
- Given that the height [tex]\( h \)[/tex] is double the side length [tex]\( s \)[/tex], we have [tex]\( h = 2s \)[/tex].
We'll solve for [tex]\( s \)[/tex]:
#### Solution:
- Side Length for Hexagonal base: [tex]\( 5 \)[/tex] cm
- Height for Hexagonal base: [tex]\( 11 \)[/tex] cm
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
Shape of Base & Side Length & Height \\
\hline
square & 7 \, \text{cm} & 14 \, \text{cm} \\
\hline
regular hexagon & 5 \, \text{cm} & 11 \, \text{cm} \\
\hline
\end{tabular}
\][/tex]
