9

First thing first, we have

$${\left(\cosh \frac{3a}{2}\right)}^2=\frac{2025}{2}$$

Since $\cosh (x)\ge 1$, we only keep the positive value

$$\cosh \frac{3a}{2}=\frac{45\sqrt{2}}{2}$$

Let $\theta =\frac{a}{2}$, then $\frac{3a}{2}=3\theta$. Since we have

$$\cosh (3\theta )=4{\cosh }^3\theta -3\cosh \theta$$

Substitute $\cosh (3\theta )=\frac{45\sqrt{2}}{2}$ into the equation, we get

$$4{\cosh }^3\theta -3\cosh \theta =\frac{45\sqrt{2}}{2}$$

let $x=\frac{\cosh \theta }{\sqrt{2}}$, we have

$$16x^3-6x-45=0$$

The solution is $x=\frac{3}{2}$, therefore

$$\cosh \theta =\frac{3\sqrt{2}}{2}$$

Also we know that

$$\cosh (a)=\cosh (2\theta )=2{\cosh }^2\theta -1$$

Substitute $\cosh \theta =\frac{3\sqrt{2}}{2}$ into the equation, we get the final answer

$$\cosh (a)=2{\left(\frac{3\sqrt{2}}{2}\right)}^2-1=\boxed{8}$$