Inspired by Hung Woei Neoh

We know that ( Sine Rule ),

$\implies \dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$

Let,

$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}=k$

$\begin{cases} a=k \ \sin A \\ b=k \ \sin B \\ c=k \ \sin C\end{cases}$

$\implies a\cos A+b\cos B+c\cos C$

$=k \ \sin A \ \cos A+k \ \sin B \ \cos B+k \ sin C \ \cos C$

$=\dfrac{k}{2}\left(\color{#3D99F6}{\sin 2A+\sin 2B+\sin 2C}\right)$

$=\dfrac{k}{2}\left(\color{#3D99F6}{4\sin A \ \sin B \ \sin C}\right)$

$=2a \ \sin B \ \sin C$

$=2a×\dfrac{2\Delta}{ac}×\dfrac{2\Delta}{ab} \ \ \ \ \ \left[ \Delta=\frac{1}{2} ab \ \sin C=\dfrac{1}{2}ca \ \sin B\right]$

$=\dfrac{8{\Delta}^{2}}{abc}$

Now, using Herons's formula

$\implies \Delta=\sqrt{s(s-a)(s-b)(s-c)}$

$\Delta=\sqrt{12\color{#20A900}{(12-a)(12-b)(12-c)}}$

$\Delta=\sqrt{12\color{#20A900}{(12)}}$

${\Delta}^{2}={12}^2=144$

Now, using Vieta's formula.

$\Rightarrow abc=420$

$\implies a\cos A+b\cos B+c\cos C=\dfrac{8×144}{420}=\dfrac{96}{35}$

$\therefore p+q+10=96+35+10=\boxed{141}$

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Note: $\Delta=\text{Area of triangle}$ .

By Vieta's formula , we have $\begin{cases} a+b+c=24 \\ ab+bc+ca=180 \\ abc = 420 \end{cases}$

By cosine rule, we have:

$\begin{aligned} a^2 & = b^2+c^2 - 2bc \cos A \\ \implies \cos A & = \frac {b^2+c^2-a^2}{2bc} \\ & = \frac {a^2+b^2+c^2-2a^2}{\frac {2abc}a} \\ & = \frac {(a+b+c)^2-2(ab+bc+ca)-2a^2}{\frac {2abc}a} \\ & = \frac {(108-a^2)a}{420} \\ \implies a \cos A & = \frac {108a^2-a^4}{420} \\ \sum_{\text{cyc}} a \cos A & = \sum_{\text{cyc}} \frac {108a^2-a^4}{420} \\ & = \frac {108(\color{#3D99F6}{216})-\color{#3D99F6}{22176}}{420} & \small \color{#3D99F6}{\text{See Note.}} \\ & = \frac{96}{35} \end{aligned}$

$\implies p+q+10 = 96+35+10 = \boxed{141}$

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$\color{#3D99F6}{\text{Note:}}$

$\begin{aligned} a^2+b^2+c^2 & = (a+b+c)^2 - 2(ab+bc+ca) \\ & = 24^2 - 2(180) = \color{#3D99F6}{216} \\ a^3+b^3+c^3 & = (a+b+c)(a^2+b^2+c^2) - (ab+bc+ca)(a+b+c) + 3abc \\ & = 24(216) - 180(24) + 3(420) = 2124 \\ a^4+b^4+c^4 & = (a+b+c)(a^3+b^3+c^3) - (ab+bc+ca)(a^2+b^2+c^2) + abc(a+b+c) \\ & = 24(2124) - 180(216) + 420(24) = \color{#3D99F6}{22176} \end{aligned}$