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 3 cents (Posted on 2014-08-29)
Place two pennies on a table each touching a third, but not each other. The centers of the three form a certain angle.

For what angle will the area of the convex hull of this shape be maximized?

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 Solution | Comment 1 of 3

`Let A, B, and C be the centers of the three`
`pennies with the penny withe center A touching`
`the other two. Let 2x be the measure of angle`
`BAC and a the common radius of the pennies.`
`The area of the convex hull is`
`  A(x) = Area(triangle ABC) + `
`         a*Perimeter(triangle ABC) +`
`         Pi*a^2`
`       = 2*a^2*sin(2x) + a*[4*a + 4*a*sin(x)] +`
`         Pi*a^2`
`  A'(x) = 4*a^2*cos(2x) + 4*a^2*cos(x)`
`  A"(x) = -8*a^2*sin(2x) - 4*a^2*sin(x)`
`  A'(x) = 0 implies cos(2x) + cos(x) = 0`
`            implies 2*cos(2x)^2 + cos(x) - 1 = 0`
`            implies cos(x) = 1/2 or -1`
`            implies 2x = 120 or 360 degrees`
`      only 2x = 120 degrees makes sense`
`  A"(60) = -8*a^2*sin(120) - 4*a^2*sin(60)`
`           is less than 0`
`Therefore, the A(x) is maximized when`
`           angle BAC = 120 degrees.`
`QED`

 Posted by Bractals on 2014-08-30 02:43:17

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