Answer
$\frac{-10.2+6\sqrt{-{{A}^{2}}+13A-39.36}}{A-6.5}$.
Work Step by Step
$A=6.5-\frac{20.4t}{{{t}^{2}}+36}$
Multiply $\left( {{t}^{2}}+36 \right)$ on both sides of the equation $A=6.5-\frac{20.4t}{{{t}^{2}}+36}$,
$\begin{align}
& A=6.5-\frac{20.4t}{{{t}^{2}}+36} \\
& A\times \left( {{t}^{2}}+36 \right)=\left( 6.5-\frac{20.4t}{{{t}^{2}}+36} \right)\times \left( {{t}^{2}}+36 \right) \\
& A{{t}^{2}}+36A=\left( {{t}^{2}}+36 \right)\times \left( 6.5 \right)-\left( {{t}^{2}}+36 \right)\left( \frac{20.4t}{{{t}^{2}}+36} \right) \\
& A{{t}^{2}}+36A=6.5{{t}^{2}}+234-20.4t
\end{align}$
This gives:
$\begin{align}
& A{{t}^{2}}-6.5{{t}^{2}}+20.4t+36A-234=0 \\
& \left( A-6.5 \right){{t}^{2}}+20.4t+\left( 36A-234 \right)=0
\end{align}$
Using the quadratic formula $x=-b\pm \frac{\sqrt{{{b}^{2}}-4ac}}{2a}$ to solve for t,
$\begin{align}
& t=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \\
& =\frac{-20.4\pm \sqrt{{{\left( 20.4 \right)}^{2}}-4\left( A-6.5 \right)\left( 36A-234 \right)}}{2\left( A-6.5 \right)} \\
& =\frac{-20.4\pm \sqrt{416.16-144{{A}^{2}}+1872A-6084}}{2\left( A-6.5 \right)} \\
& =\frac{-20.4\pm \sqrt{-144{{A}^{2}}+1872A-5667.84}}{2\left( A-6.5 \right)}
\end{align}$
Simplify more,
$\begin{align}
& t=\frac{-20.4\pm \sqrt{144\left( -{{A}^{2}}+13A-39.36 \right)}}{2\left( A-6.5 \right)} \\
& =\frac{-20.4\pm 12\sqrt{-{{A}^{2}}+13A-39.36}}{2\left( A-6.5 \right)} \\
& =\frac{2\left( -10.2\pm 6\sqrt{-{{A}^{2}}+13A-39.36} \right)}{2\left( A-6.5 \right)} \\
& =\frac{-10.2\pm 6\sqrt{-{{A}^{2}}+13A-39.36}}{A-6.5}
\end{align}$
Since we only consider the positive answer: $t=\frac{-10.2+6\sqrt{-{{A}^{2}}+13A-39.36}}{A-6.5}$.
