Answer:
Step-by-step explanation:
To find the volume of a triangular prism, we use the formula:
�
=
Base Area
×
Height
V=Base Area×Height
In this case, the base of the triangular prism is a triangle with dimensions 7 yards, 10 yards, and 6 yards.
First, we need to find the area of the base triangle using Heron's formula:
Area
=
�
(
�
−
�
)
(
�
−
�
)
(
�
−
�
)
Area=
s(s−a)(s−b)(s−c)
where
�
s is the semi-perimeter of the triangle, and
�
,
�
,
a,b, and
�
c are the lengths of its sides.
The semi-perimeter
�
s is given by:
�
=
�
+
�
+
�
2
s=
2
a+b+c
Given that the lengths of the sides are
�
=
7
a=7 yards,
�
=
10
b=10 yards, and
�
=
6
c=6 yards, we can calculate
�
s:
�
=
7
+
10
+
6
2
=
23
2
=
11.5
s=
2
7+10+6
=
2
23
=11.5
Now, we can use Heron's formula to find the area of the base:
Area
=
11.5
(
11.5
−
7
)
(
11.5
−
10
)
(
11.5
−
6
)
Area=
11.5(11.5−7)(11.5−10)(11.5−6)
Area
=
11.5
×
4.5
×
1.5
×
5.5
Area=
11.5×4.5×1.5×5.5
Area
=
557.8125
Area=
557.8125
Area
≈
23.6
square yards
Area≈23.6 square yards
Now, we multiply the area of the base by the height of the prism, which is 8 yards:
�
=
23.6
×
8
V=23.6×8
�
=
188.8
cubic yards
V=188.8 cubic yards
Therefore, the volume of the triangular prism is approximately
188.8
188.8 cubic yards.
