Given:
The volume of the triangular prism = 27 cubic units
To find the volume of the prism if each side is dilated by [tex]\frac{1}{3}[/tex]
Formula
The volume of a triangular prism
[tex]V = \frac{1}{2} ach[/tex]
where, a be the height of the triangle
c be the base of the triangle
h be the height of the prism
Let us take after dilation the we get,
[tex]a \rightarrow \frac{1}{3} a[/tex]
[tex]c \rightarrow \frac{1}{3} c[/tex] and
[tex]h \rightarrow \frac{1}{3} h[/tex]
So,
After dilation the volume will be
[tex]V_{1} =\frac{1}{2} (\frac{a}{3} )(\frac{c}{3} )(\frac{h}{3} )[/tex]
or, [tex]V_{1} =\frac{1}{9} (\frac{1}{2} ach)[/tex]
or, [tex]V_{1}=\frac{1}{9} (27)[/tex] [Since, [tex]V = \frac{1}{2} ach = 27[/tex]]
or, [tex]V_{1} = 3[/tex]
Hence,
After dilation the volume of the prism will be 3 cubic unit.
