Use the image to answer the question.
An illustration shows a triangle with sides measuring 16 meters, 14 meters, and 8 meters. A dashed perpendicular line, from the side measuring 16 meters to the opposite angle, measures 7 meters. A right angle symbol is shown to the left of the perpendicular line.
A prism and pyramid have congruent triangular bases. If their heights are both 15 m, what is the volume of each shape?
To find the volume of a prism, we use the formula: V = base area x height. Since the base of the prism is a triangle with sides 16m, 14m, and 8m, we first need to calculate the area of the base. We can use Heron's formula to find the area:
s = (16 + 14 + 8) / 2 = 19 meters
Area = sqrt(19(19-16)(19-14)(19-8)) = sqrt(19*3*5*11) = sqrt(3135) ≈ 56 meters
Now, we can find the volume of the prism:
V = 56m x 15m = 840 m^3
Since the pyramid also has a triangular base with the same dimensions, its volume will be 1/3 of the prism's volume:
V(pyramid) = 1/3 x 840 m^3 = 280 m^3
Therefore, the volume of the prism is 840 cubic meters, and the volume of the pyramid is 280 cubic meters.