check the picture below.
so we can use the pythagorean theorem to get the dashed line of the road,
[tex] \bf c=\sqrt{a^2+b^2}\implies c=\sqrt{5^2+9^2}\implies \stackrel{\textit{miles on the map}}{c=\sqrt{106}} [/tex]
we know that Sam doing 30mph, can do that road in 45 minutes, in actual length, well, 45 minutes is 3/4 of an hour, so
[tex] \bf \stackrel{45~mins}{\cfrac{3}{4}\underline{hr}}\cdot \cfrac{30miles}{\underline{hr}}\implies \cfrac{90miles}{4}\implies \cfrac{45}{2}~miles [/tex]
so, that's how many actual miles the road really is, so if we put that in actual : map ratio, we'd get
[tex] \bf actual:map\qquad \cfrac{actual}{map}\implies \cfrac{\quad \frac{45}{2}\quad }{\sqrt{106}}\implies \cfrac{45}{2\sqrt{106}}\textit{ miles per cm} [/tex]
and you know what that is.
