3470.61 – Barber Pole

The revolving portion of the splendid barber pole in the figure is a cylinder 4 feet tall and 6 inches in radius. The stripe spirals around the cylinder making exactly 8 complete turns from bottom to top.

Ignoring the width of the stripe, how long is it?

Solution

You can make a great model of this problem by drawing a right triangle with a red hypotenuse and wrapping it around a pencil. Tape the 48” side to the pencil first. Unwrap it in front of the class when they’ve gotten stuck, watch their eyes widen.

Meanwhile, the triangle in the figure below makes clear that the stripe (red or black, or any other color, actually) is the hypotenuse of a right triangle whose base is a circle of radius 6" unwound 8 times. That makes the base 2π6⋅8=96π≈302".2 \pi 6 \cdot 8 = 96 \pi \approx 302". By the Pythagorean theorem, the stripe has length,

stripe =3022+482≈306" or 12.5′. \text{stripe } = \sqrt{302^2 + 48^2} \approx 306" \text{ or } 12.5'.