The radii of an ellipse are a and b (a>b). Find the inserted rectangles that have:
the maximal area
the maximal circumference
If you did not get stationary points for the minima, explain why!
Get those minimal points!
Hint
Parametric form of an ellipse:
(1.3.3.23)
Parts 1-6
Solution of question 1.
The sides of an inserted rectangle are
2x and 2y ,
where x and y are defined by (1.3.3.23) for .
The reason for using the first quadrant is that only there both x and y are non negative.
The area of the rectangle is
its derivative is
which vanishes for
The second derivative is negative:
and therefore the maximal area is
Parts 7-12
Solution of question 2.
The circumference of the rectangle is
its derivative is
which vanishes for
and in terms of sin and cos
The second derivative is negative:
and therefore the maximal circumference is
Parts 13-14
Solution of question 3
Obviously the minimal area is zero and the minimal circumference is 4b .
The limitation of we used, put the minima at the end points of this interval and therefore it was impossible to get stationary points there.
Similar situation was encountered at an other exercise.
By substituting , one obtains accordingly:
and
Score
Correct solution of question 1 gives credit of 3 points.
Correct solution of question 2 gives credit of 3 points.
Correct solution of question 3 gives credit of 4 points.
By parts:
Parts 1,...,12 are worth half a point each.
Parts 13,14 are worth 2 points each.
If the total of the score is not an integer, add half a point.