Let the volumes of $ A, B,$ and $C$ be $3x, 4x$ and $7x.$

Weights of the same volume of the metals $ A, B,$ and $C$ are in the ratio $5:2:6.$

So, the ratio of the weights in the overall alloy.

$A:B:C = (3x \times5) : (4x \times 2) : (7x \times 6)$

$ \Rightarrow A:B:C = 15 : 8 : 42 $

$\therefore$ The weight of metal $C = 130 \times \frac{42}{65} = 84 \; \text{kg}.$

$\textbf{Short Method}:$

$\begin{array}{} \text{} & A & B & C \\ \text{Volume:} & 3: &4: & 6 \\ \text{Weight:} & 5: & 2: & 6(1 \;\text{litre}) \\ \text{Weight:} & 15: & 8: & 42 \end{array}$

Now,

- $ 65 \longrightarrow 130$
- $1 \longrightarrow 2 $
- $ 42 \longrightarrow 84 \; \text{kg}$

Correct Answer$: \text{D}$