Answer: C

Solution:

The top is a ridge of length $7,$ centered above the base at some height $h.$ Its endpoints sit above $(3, 4)$ and $(10, 4)$ of the $13 \times 8$ base. A slant edge to a base corner has length $\sqrt{3^2 + 4^2 + h^2} = 13,$ so $h = 12.$

At height $z,$ the horizontal cross-section is a rectangle measuring $\left(13 - \dfrac{z}{2}\right)$ by $\left(8 - \dfrac{2z}{3}\right).$ At $z = 0$ its area is $104$; at $z = 6$ it is $10 \cdot 4 = 40$; at $z = 12$ the ridge has area $0.$

By the prismatoid formula, $$\begin{aligned} V &= \frac{12}{6}\left(104 + 4 \cdot 40 + 0\right) \\ &= 2(264) = 528. \end{aligned}$$

Thus, the correct answer is C.