1. We can identify the \(\text{base} = 4\;[ft]\), \(\text{height} = 3\;[ft]\) and \(\text{length} = 7\;[ft]\).
We can plug the given values into the formula for the volume of a triangular prism:
\(\text{Volume} = \cfrac{1}{2}bhl\)
\(\text{Volume} = \cfrac{1}{2}(4\;[ft])(3\;[ft])(7\;[ft])\)
\(\text{Volume} = 42\;[ft^3]\)
Therefore, we can determine that the Volume of the tent is \(42\;[ft³]\).
2. We need to identify the total Surface Area of the tent by adding up the individual sides.
We can start by calculating the Surface Area of the floor:
Floor: \((Base)(Length)\)
Floor: \((4ft)(7ft) = 28ft^2\)
We can then calculate the Surface Areas of the triangular faces:
Triangular Faces: \((2)(\cfrac{1}{2})(\text{Base}) (\text{Height})\)
Triangular Faces: \((2)(\cfrac{1}{2})(3\;[ft])(4\;[ft])\)
Triangular Faces: \(= 12ft^2\)
Finally, we can find the Surface Area of the rectangular sides. We first need to calculate the hypoteneuse to determine their widths:
\(c^2 = a^2 + b^2\)
\(c^2 = 3^2 + 2^2\)
\(\sqrt{c^2} = \sqrt{14}\)
\(c = 3.74\)
Rectangular Sides: \((2)(\text{Length})(\text{Width})\)
Rectangular Sides: \((2)(7\;[ft])(3.74\;[ft]) = 52.36\;[ft^2]\)
\(\text{SA} = 28\;[ft^2] + 12\;[ft^2] + 52.36\;[ft^2]\)
\(\text{SA} = 92.36\;[ft^2]\)
Therefore, we can determine that we need \(92.36\;[ft²]\) of wood to make the tent.