#### Answer

$$ \ln (5+\sqrt{26})$$

#### Work Step by Step

Since
\begin{align*}
s&=\int_{a}^{b} \sqrt{1+\left(y^{\prime}\right)^{2}} d x\\
&=\int_{-a}^{a} \sqrt{1+(\sinh x)^{2}} d x\\
&=\int_{-a}^{a} \sqrt{1+\sinh ^{2} x} d x\\
&=\int_{-a}^{a} \sqrt{\cosh ^{2} x} d x\\
&=\int_{-a}^{a} \cosh x d x\\
&=[\sinh x]_{-a}^{a}\\
&=2 \sinh a
\end{align*}
Setting this expression equal to 10 and solving for $a$ yields $$a=\sinh ^{-1}(5)=\ln (5+\sqrt{26})$$