A spiral staircase turns 270 degrees as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth.
The easiest way to see this is to imagine that we can "unfold" the spiral to create a rectangle
The bottom side of the rectangle is just the arc length of a circle of a radius of 3 ft turned through 270°= (3/4) of a whole revolution....in other words.....(3/4) (2pi) (3) = (9/2)pi ft = 4.5 pi ft
The side of the rectangle will be rise of the staircase = 10ft
And the diagonal of the rectangle will be the length of the handrail
We can use the Pythagorean Theorem to solve this
handrail length = √ [ (4.5 pi)^2 + 10^2 ] ≈ 17.3 ft