# I Wish The Car Stayed Still

Calculus Level 4

A ladybug is climbing on a Volkswagen Beetle (a type of car). In its starting position, the surface of the car is represented by the unit semicircle $$x^{2} + y^{2} = 1, y \geqslant 0$$ in the $$xy$$-plane. The $$x$$-axis is the road. At time $$t = 0$$ the ladybug starts at the front bumper $$(1, 0)$$, and crawls counterclockwise on the surface of the car at a unit speed relative to the car. At the same time, the car moves to the right at speed 10.

If the speed of the bug at $$t = \dfrac{\pi}{4}$$ can be represented as $$S$$, find $$\left \lfloor 10S \right \rfloor$$.

###### Credit: Problem information is adapted from MIT OCW.
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