Final answer:
To calculate the volume of the pentagonal prism, find the area of the pentagon cross-section using its formula. Substitute the cross-sectional area and side length into the volume formula.
Step-by-step explanation:
To calculate the volume of the pentagonal prism, we first need to find the area of the pentagon cross-section. The area of a pentagon is given by the formula:
A = (1/4) * sqrt(5 * (5 + 2 * sqrt(5))) * s^2,
where s is the length of a side of the pentagon. Given that the cross-sectional area is 27 cm², we can solve for s and substitute it into the formula for the volume of the prism:
V = A * h
= (1/4) * sqrt(5 * (5 + 2 * sqrt(5))) * s^2 * 3 cm
= 27 * 3 = 81 cm³.