Let's do some calculus! (36)

Calculus Level 4

A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a straight trajectory at \(1 \ {m}/{s}\).

Suddenly someone turns on (spins) the turntable and it spins at \(30 \ rpm\). Consider the set \(S\) of points the mouse can reach in his car within \(1 \ \text{second}\) after the turntable is set in motion. (Just for clarity, the arrows in the figure below represent two possible paths the mouse can take.)

What is the area of \(S\), in square meter(s)?

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