The sum of the radius of the base and the height of a solid cylinder is 37 metres.

Let r and h be the base radius and the height of the solid cylinder, respectively.

We have,

$(r+h)=37 \mathrm{~m}$

As, the total surface area of the cylinder $=1628 \mathrm{~m}^{2}$

$\Rightarrow 2 \pi r(r+h)=1628$

$\Rightarrow 2 \times \frac{22}{7} \times r \times 37=1628$

$\Rightarrow r=\frac{1628 \times 7}{2 \times 22 \times 37}$

$\Rightarrow r=7 \mathrm{~m}$

So, $h=(37-7)=30 \mathrm{~m}$

Now, the volume of the solid cylinder $=\pi r^{2} h$

$=\frac{22}{7} \times 7 \times 7 \times 30$

$=4620 \mathrm{~m}^{3}$