Answer: 410 square km
Step-by-step explanation:
The area of each of the triangles with their base on the 10 km side is [tex]A=\frac{1}{2}(10)(13.4)[/tex], so their combined area is [tex](10)(13.4)=134[/tex].
The area of each of the triangles with their base on the 12 km side is [tex]\frac{1}{2}(12)(13)[/tex], so their combined area is [tex](12)(13)=156[/tex].
The area of the bottom rectangular face is [tex]10(12)=120[/tex].
So, the total surface area is [tex]120+156+134=\boxed{410 \text{ km}^{2}}[/tex]
