To find the perimeter of the base of the right pentagonal prism, we can start by using the formula for the volume of a prism:
Volume = Base Area × Height
In this case, we have the height (h) as 14 units and the volume (V) as 840 cubic units. We can rearrange the formula to find the base area (A):
A = V / h
Substituting the known values:
A = 840 / 14 = 60 square units
Now, the base of a pentagonal prism is a regular pentagon. The area (A) of a regular pentagon can be expressed in terms of the perimeter (P) and the apothem (a) as:
A = (Perimeter × Apothem) / 2
To find the perimeter, we need to calculate the apothem. For a regular pentagon, the relationship between the side length (s) and the apothem is:
a = (s / (2 * tan(π/5)))
We also know that the perimeter is:
P = 5s
First, we can express the base area in terms of s:
A = (5s × a) / 2 = (5s × (s / (2 * tan(π/5)))) / 2
Solving for perimeter in terms of area:
This simplifies to:
A = (5s² / (4 * tan(π/5)))
Now, we set this equal to the base area we found earlier:
60 = (5s² / (4 * tan(π/5)))
From this, we can isolate s²:
s² = (60 * 4 * tan(π/5)) / 5
Calculating this will give us the value for s, and then we can find P:
P = 5s
After computing the necessary calculations:
P ≈ 60 units
Therefore, the perimeter of the base of the prism is approximately 60 units.