Playing with Pencils

Calculus Level 5

Betsy is drawing with a mechanical pencil. She ends up drawing the closed figure shown above. All units are in inches.

This figure is given by the formula sin(x) from 0 to 2$$\pi$$. Betsy desides to shade in the parts that are shaded in on the graph. She then decides to trace the outer ends of the figure with a thickness of 1/50 of an inch without lifting her pencil.

She uses 1 piece of graphite after drawing 80.56 $$\text{inches} ^2$$. Let a be the number of these figures (with the shaded and outlined areas) that she can draw with this 1 piece of graphite. Find a+1.

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