10. A box has the shape of a right prism with a regular hexagon for its base. The sides in the base are 10 cm long. The box is 20 cm high and the area of the base is 520 cm².
a) How many faces of each type does the box without the lid have?
b) Sketch the faces of the box. Write the dimensions on the faces.
c) Find the surface area of the box without the lid.
a) The box without the lid has 5 rectangular faces and one regular hexagonal face.
b) The sketch of the box would look like this:
\begin{equation}
\begin{split}
&\text{_________}\\
& / / \\
& /____/ / \\
&| |\\
&| |\\
&| |\\
&| |\\
&| |\\
&| |\\
&| |\\
&| |\\
&| |\\
& \------------/\\
\end{split}
\end{equation}
The dimensions on the faces are as follows:
- Rectangular faces: 10 cm (width) x 20 cm (height)
- Hexagonal face: side length of 10 cm
c) The surface area of the box without the lid is the sum of the areas of its faces. We have 5 rectangular faces with an area of 10 cm x 20 cm = 200 cm² each, and one hexagonal face with area 520 cm². So the total surface area is:
5 x 200 cm² + 520 cm² = 1520 cm²