All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes
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    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Parallel Rectangles Comment 2 of 2 |
This time I have all three rectangles parallel to each other but not joined in one big brick.  This time all three rectangle centers will remain on the x-axis.

Like last time, my ellipse is x^2/9+y^2=1.  The 'vertical' sides of the rectangles tangent to the ellipse are ((cos t)/3)*x + (sin t)*y = +/-1.  The 'vertical' sides of the center rectangle are ((cos t)/3)*x + (sin t)*y = +/-1/3.  (I use the word vertical loosely here to represent the sides/lines I expect to be closer to vertical slope than horizontal slope in the solution.  At t=0 these are true vertical lines.)

The horizontal lines of the center rectangle are (sin t)*x - ((cos t)/3)*y = 1.

At t=0 the vertecies are far outside the ellipse, so the answer is a matter of finding the angle when a vertex lies on the ellipse.  Again, I opted for the spreadsheet to get a value of t=0.24261262721
rad.

The vertecies of the center rectangle are at (1.65021370961, -0.83511773705); (-0.32203403613, 1.82124160992); 
(-1.65021370961, 0.83511773705); (0.32203403613, -1.82124160992)

The sides of each rectangle are 3.30847489802 by 1.65423744901, for an area of 5.47300307540 for each rectangle.  This time the ratio of ellipse to total rectangle area is 0.57401624123, an improvement over the simple big rectangle.




  Posted by Brian Smith on 2016-12-16 16:01:39
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (14)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information