jmc

The handrail encases a right circular cylinder with radius 3 feet and height 10 feet. Its lateral area is a rectangle with height 10 feet and width equal to its base circumference, or $2\pi \cdot 3=6\pi$ feet. A staircase that turns $360^{\circ }$ would, when unrolled and lain flat, span the diagonal of this rectangle. However, our staircase does not make a full turn, so it spans a rectangle with a shorter width.

A $270^{\circ }$ sector of a circle with radius 3 has arc length $\frac{270^{\circ }}{360^{\circ }}\cdot 2\pi \cdot 3=4.5\pi$. Thus, when unrolled and lain flat, our handrail spans the diagonal of a rectangle with height 10 feet and width $4.5\pi$ feet. Our handrail has length $\sqrt{10^2+(4.5\pi {)}^2}\approx 17.317$ feet. To the nearest tenth, this value is $\boxed{17.3}$ feet.