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?
(1 point)
The volume of the prism is
m3.
The volume of the pyramid is
m3.
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To find the volume of the prism, we use the formula: Volume = Base Area x Height. Since the base of the prism is a triangle with sides measuring 16, 14, and 8 meters, we can use Heron's formula to find the area of the base.
First, find the semiperimeter:
s = (16 + 14 + 8) / 2 = 19 meters
Now, find the area of the triangle using Heron's formula:
Area = sqrt[s(s-16)(s-14)(s-8)] = sqrt[19(3)(5)(11)] = sqrt(3135) = 56.03 m^2
Now, calculate the volume of the prism:
Volume = Base Area x Height = 56.03 m^2 x 15 m = 840.45 m^3
Therefore, the volume of the prism is 840.45 m^3.
The volume of a pyramid is given by the formula: Volume = (1/3) x Base Area x Height.
Since the base of the pyramid is congruent to the base of the prism, the base area is also 56.03 m^2.
Now, calculate the volume of the pyramid:
Volume = (1/3) x 56.03 m^2 x 15 m = 280.23 m^3
Therefore, the volume of the pyramid is 280.23 m^3.