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A two dimensional silo shaped figure is formed by placing a semi-circle of diameter 1 on top of a unit square, with the diameter coinciding with the top of the square. How do we find the radius of the smallest circle that contains this silo?

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Draw the line of length 1.5 that cuts both the square and the half circle into two identical pieces. (Starting from the middle of the base of the square, go straight up) Notice that the center of the larger circle must lie somewhere on this line. Say that the center point is distance $x$ from the base of the square, and that this circle has radius $r$. Than, the distance from the center point to the corner of the square should be $r$, so by Pythagoras we find $$r^2 = x^2 +(1/2)^2$$. Now, the distance to the top of the half circle from this center point is $1.5-x$ and this should also be the radius. Hence $$1.5-x=r.$$ Thus you have two equations and two unknowns, which you can solve from here.

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