1. **State the problem:** Find the volume of the given triangular prism with dimensions: height of the triangle base $=5$, base edges $7$ and $8$. 2. **Formula for volume of a prism:** $$\text{Volume} = \text{Base Area} \times \text{Length}$$ where the base area is the area of the triangular base and the length is the prism's length perpendicular to the base. 3. **Calculate the base area:** The base is a triangle with sides $7$, $8$, and height $5$ (height corresponds to the vertical height of the triangle). Area of triangle = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \times 5 = \frac{35}{2} = 17.5$ 4. **Calculate the volume:** The prism length is $8$ (the edge perpendicular to the triangular base). $$\text{Volume} = 17.5 \times 8 = 140$$ 5. **Answer:** The volume of the prism is $140$ units$^3$. Therefore, the correct choice is **C. 140 units^3**.