Answer:
[tex]576 \, \textsf{cm}^2[/tex]
Step-by-step explanation:
The surface area ([tex]SA[/tex]) of a triangular prism is the sum of the areas of its three rectangular faces and the areas of its two triangular faces.
Let's denote [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex] as the sides of the triangular base, [tex]h[/tex] as the height of the triangular base, and [tex]L[/tex] as the length of the prism.
The formula for the surface area of a triangular prism is given by:
[tex] SA = 2 \times \textsf{Area of Triangular Base} + \textsf{Perimeter of Triangular Base} \times \textsf{Length of Prism} [/tex]
Now, let's calculate the components of this formula.
Area of Triangular Base ([tex]A_{\textsf{base}})[/tex]:
[tex] A_{\textsf{base}} = \dfrac{1}{2} \times \textsf{Base} \times \textsf{Height} [/tex]
[tex] A_{\textsf{base}} = \dfrac{1}{2} \times 12 \times 8 [/tex]
[tex] A_{\textsf{base}} = 48 \, \textsf{cm}^2 [/tex]
Perimeter of Triangular Base ([tex]P_{\textsf{base}})[/tex]:
[tex] P_{\textsf{base}} = a + b + c [/tex]
[tex] P_{\textsf{base}} = 10 + 12 + 10 [/tex]
[tex] P_{\textsf{base}} = 32 \, \textsf{cm} [/tex]
Length of Prism ([tex]L[/tex]):
Given as [tex]L = 15 \, \textsf{cm}[/tex].
Now, substitute these values into the surface area formula:
[tex] SA = 2 \times A_{\textsf{base}} + P_{\textsf{base}} \times L [/tex]
[tex] SA = 2 \times 48 + 32 \times 15 [/tex]
[tex] SA = 96 + 480 [/tex]
[tex] SA = 576 \, \textsf{cm}^2 [/tex]
Therefore, the surface area of the triangular prism is [tex]576 \, \textsf{cm}^2[/tex].
