Let's solve each question step by step:
### Question 1: Volume of the Box with Height 7 inches
1. Determine the Area of the Octagonal Base:
- The octagonal base is a regular octagon with each side of 1 inch.
- The area of a regular octagon can be calculated using the formula for polygons, however, it has already been given that for this particular octagon, the area is approximately [tex]\( 2.828 \, \text{in}^2 \)[/tex].
2. Calculate the Volume:
- The volume [tex]\( V \)[/tex] of a prism is given by the product of the area of its base [tex]\( A \)[/tex] and its height [tex]\( h \)[/tex].
- Given [tex]\( A = 2.828 \, \text{in}^2 \)[/tex] and [tex]\( h = 7 \, \text{in} \)[/tex]:
[tex]\[
V = A \times h = 2.828 \, \text{in}^2 \times 7 \, \text{in} = 19.799 \, \text{in}^3
\][/tex]
So, the volume of the box is [tex]\( 19.799 \, \text{in}^3 \)[/tex].
### Question 2: Height Given Volume 123 in³
1. Determine the Area of the Octagonal Base:
- Again, we use the previously calculated area of the base, which is [tex]\( 2.828 \, \text{in}^2 \)[/tex].
2. Calculate the Height:
- Given the volume [tex]\( V = 123 \, \text{in}^3 \)[/tex] and the area of the base [tex]\( A = 2.828 \, \text{in}^2 \)[/tex], the height [tex]\( h \)[/tex] can be found using the formula:
[tex]\[
h = \frac{V}{A} = \frac{123 \, \text{in}^3}{2.828 \, \text{in}^2} = 43.487 \, \text{in}
\][/tex]
So, the height of the box is [tex]\( 43.487 \, \text{in} \)[/tex].
### Summary
1. Volume of the box with height 7 inches: [tex]\( 19.799 \, \text{in}^3 \)[/tex]
2. Height of the box given a volume of 123 in³: [tex]\( 43.487 \, \text{in} \)[/tex]
These calculations provide the detailed steps to answer both questions accurately.
