To find the dimensions of the toy pyramid with maximum material usage of 250 square centimeters and the height being double the side length, we consider both a square base and a regular hexagonal base.
### Square Base Pyramid:
1. Formula for total surface area: The total surface area of a square pyramid includes the base area and the four triangular faces.
[tex]\[
\text{Total Surface Area} = \text{base area} + 4 \times \text{triangle area}
\][/tex]
2. Base area: If the side length is [tex]\(a\)[/tex],
[tex]\[
\text{base area} = a^2
\][/tex]
3. Triangle area: Each triangular face has an area,
[tex]\[
\text{triangle area} = \frac{1}{2} \times a \times \text{height}
\][/tex]
with the height of the pyramid being [tex]\(2a\)[/tex], thus,
[tex]\[
\text{triangle area} = \frac{1}{2} \times a \times (2a) = a^2
\][/tex]
4. Combine areas: The total surface area is,
[tex]\[
\text{Total Surface Area} = a^2 + 4 \times a^2 = 5a^2
\][/tex]
5. Set the surface area equal to maximum material:
[tex]\[
5a^2 = 250 \implies a^2 = 50 \implies a = \sqrt{50} \approx 7
\][/tex]
6. Height: Since height [tex]\(= 2a\)[/tex],
[tex]\[
\text{Height} = 2 \times 7 = 14
\][/tex]
So, for a square base:
[tex]\[
\text{Side Length} = 7 \text{ cm, Height} = 14 \text{ cm}
\][/tex]
### Hexagonal Base Pyramid:
1. Formula for total surface area: The total surface area of a hexagonal pyramid includes the base area and the six triangular faces.
[tex]\[
\text{Total Surface Area} = \text{base area} + 6 \times \text{triangle area}
\][/tex]
2. Base area: If the side length is [tex]\(a\)[/tex], then the area of a regular hexagon is,
[tex]\[
\text{base area} = \frac{3\sqrt{3}}{2} a^2
\][/tex]
3. Triangle area: Each triangular face has an area,
[tex]\[
\text{triangle area} = \frac{1}{2} \times a \times \text{height}
\][/tex]
with the height of the pyramid being [tex]\(2a\)[/tex], thus,
[tex]\[
\text{triangle area} = \frac{1}{2} \times a \times (2a) = a^2
\][/tex]
4. Combine areas: The total surface area is,
[tex]\[
\text{Total Surface Area} = \frac{3\sqrt{3}}{2} a^2 + 6a^2 = \left(\frac{3\sqrt{3}}{2} + 6\right) a^2
\][/tex]
5. Set the surface area equal to maximum material:
[tex]\[
250 = \left(\frac{3\sqrt{3}}{2} + 6\right) a^2
\][/tex]
Solving for [tex]\(a\)[/tex] (side length):
[tex]\[
a \approx 5
\][/tex]
6. Height: Since height [tex]\(= 2a\)[/tex],
[tex]\[
\text{Height} = 2 \times 5 = 11
\][/tex]
So, for a hexagonal base:
[tex]\[
\text{Side Length} = 5 \text{ cm, Height} = 11 \text{ cm}
\][/tex]
Thus, the correct dimensions to the nearest square centimeter are:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline Shape of Base & Side Length & Height \\
\hline square & 7 cm & 14 cm \\
\hline regular hexagon & 5 cm & 11 cm \\
\hline
\end{tabular}
\][/tex]
