To find the maximum dimensions for a toy pyramid with a given maximum material of 250 square centimeters, we will need to determine the side length and height for both square and hexagonal bases.
### Square Base Pyramid:
1. Side Length:
The side length of the square base is given by:
[tex]\[
a = 5 \sqrt{-\frac{5}{8} + \frac{5\sqrt{17}}{8}}
\][/tex]
2. Height:
The height of the pyramid, which is double the side length, is:
[tex]\[
h = 10 \sqrt{-\frac{5}{8} + \frac{5\sqrt{17}}{8}}
\][/tex]
Evaluating these expressions numerically, we get the following values when approximated to the nearest centimeter:
[tex]\[
a \approx 10 \, \text{cm}
\][/tex]
[tex]\[
h \approx 20 \, \text{cm}
\][/tex]
### Hexagonal Base Pyramid:
1. Side Length:
The side length of the hexagonal base is given by:
[tex]\[
b = \sqrt{\frac{1250}{33} - \frac{250\sqrt{3}}{33}}
\][/tex]
2. Height:
The height of the pyramid, which is double the side length, is:
[tex]\[
h = 2 \sqrt{\frac{1250}{33} - \frac{250\sqrt{3}}{33}}
\][/tex]
Evaluating these expressions numerically, we get the following values when approximated to the nearest centimeter:
[tex]\[
b \approx 8 \, \text{cm}
\][/tex]
[tex]\[
h \approx 16 \, \text{cm}
\][/tex]
Thus, we can fill in the table format as:
\begin{tabular}{|c|c|c|}
\hline
Shape of Base & Side Length & Height \\
\hline
square & 10 cm & 20 cm \\
\hline
regular hexagon & 8 cm & 16 cm \\
\hline
\end{tabular}
