Group Event 8
G8.1 The area of an equilateral triangle of side $A \mathrm{~cm}$ is $\sqrt{3} \mathrm{~cm}^{2}$. Find $A$.
G8.2 If $19 \times 243^{\frac{A}{5}}=b$, find $b$.
G8.3 The roots of the equation $x^{3}-173 x^{2}+339 x+513=0$ are $-1, b$ and $c$. Find $c$.
G8.4 The base of a triangular pyramid is an equilateral triangle of side $2 c \mathrm{~cm}$.
If the height of the pyramid is $\sqrt{27} \mathrm{~cm}$, and its volume is $d \mathrm{~cm}^{3}$, find $d$.
$\begin{array}{l}\frac{1}{2} \cdot A^{2} \sin 60^{\circ}=\sqrt{3} \\ \Rightarrow A=2 \\ b=19 \times\left(3^{5}\right)^{\frac{2}{5}}=171 \\ -1+171+c=\text { sum of roots }=173 \\ \Rightarrow c=3 \\ d=\frac{1}{3} \cdot \frac{1}{2} \cdot\left(6^{2} \cdot \sin 60^{\circ}\right) \cdot \sqrt{27}=27\end{array}$