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Covering an Ellipse (Posted on 2014-07-26) Difficulty: 3 of 5
Each of the lengths of three identical rectangles R1, R2 and R3 is precisely twice their respective breadths.

The respective semi-major axis and the semi-minor axis of an ellipse E are 3 and 1. If E is completely covered by R1, R2 and R3, then find the smallest dimension of each of the three rectangles.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts One big rectangle | Comment 1 of 2
I am starting with the obvious - attach all three small rectangles into one big rectangle.  So I am looking for a 2:3 rectangle to circumscribe the ellipse.

Let the ellipse be x^2/9 + y^2 = 1.  The parametric form of this ellipse is (x,y) = (3*cos t, sin t).  t is the angle measured counterclockwise starting from the positive x-axis.

Then the slope at point (x,y) is dy/dx = -x/(9y) = (-1/3)*(cot t).  The equation of the tangent line can then be expressed as ((cos t)/3)*x + (sin t)*y = 1.  The other tangent parallel to this line can be expressed by changing the 1 to a -1.

Let t1 and t2 be the angles of a pair of perpendicular tangents.  Then tan(t1) * tan(t2) = -1/9

Then the rectangle is defined by the four lines (subject to the tan product equation):
((cos t1)/3)*x + (sin t1)*y = 1
((cos t1)/3)*x + (sin t1)*y = -1
((cos t2)/3)*x + (sin t2)*y = 1
((cos t2)/3)*x + (sin t2)*y = -1

At this point I just plugged this into a spreadsheet and adjusted t1 until the sides of the rectangle were in a 2:3 ratio.  This happened at t1=0.1948803663985 rad.  Then t2=-0.512704070327 rad.

The vertecies of the rectangle were at (3.157775026473, -0.168691677879), (1.370244251303, 2.849987840636), (-3.157775026473, 0.168691677879), (-1.370244251303, -2.849987840636).

Then the three individual 1:2 rectangles would have edge lengths of 1.754116038614 by 3.508232077226.

The ratio of ellipse area to rectangle area in this case is 0.510508806209, just over half of the rectangles' area is covered by the ellipse.

  Posted by Brian Smith on 2016-12-13 12:51:48
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