The volume of a prsm is given by the area of the base multiplied by the height.
Given a pentagonal prism with a perpendicular distance of 14 units between the bases and a volume of 840 cubic units. This means that the height of the prism is 14 units.
i.e. Area of base x 14 = 840
which implies that the area of the base = 840 / 14 = 60
Assuming that the pentagonal base is regular.
Area of the pentagon base is given by
[tex]A= \frac{1}{4} \sqrt{5(5+2 \sqrt{5}) } a^2[/tex]
where a is the length of the sides.
[tex]60= \frac{1}{4} \sqrt{5(5+2 \sqrt{5}) } a^2 \\ \\ \sqrt{5(5+2 \sqrt{5}) } a^2=240 \\ \\ a^2= \frac{240}{\sqrt{5(5+2 \sqrt{5}) }} =34.8740 \\ \\ a=5.9054[/tex]
Perimeter of a pentagon is given by 5a
where a is the length of the sides.
Therefore,. the perimeter of the base of the pentagonal prism is given by 5 x 5.9054 = 29.5 units.
