Chapter 10.3, Problem 54ES

As illustrated in the accompanying figure, suppose that a rod with one end fixed at the pole of a polar coordinate system rotates counterclockwise at the constant rate of $1\text{rad}/s.$ At time $t=0$ a bug on the rod is 10 mm from the pole and is moving outward along the rod at the constant speed of $2\text{mm}/\text{s}.$

(a) Find an equation of the form $r=f\left(\theta \right)$ for path of motion of the bug, assuming that $\theta =0$ when $t=0.$

(b) Find the distance the bug travels along the path in part (a) during the first $5\text{s}.$ Round your answer to the nearest tenth of a millimeter.