Find the ellipse of maximum area that can be completely covered by two unit circles.
The area of an ellipse equals pi*a*b, where a and b are the major and minor semi-axes. Maximizing a*b will maximize pi*a*b.
If x is half the distance between the centers of the circles,
b^2 + x^2 = 1
a = 1 + x
b = sqrt(1 - x^2)
a*b = (1 - x^2)^(1/2) * (1 + x)
Its derivative is
(1 - x^2)^(1/2) + (1 + x) * ((1 - x^2)^(-1/2) / 2) * (-2*x)
Wolfram Alpha finds x = 1/2 is where this derivative is zero.
The major semi-axis is 3/2 and the minor semi-axis is sqrt(3/4) = sqrt(3)/2.
As a check:
Wolfram Alpha also is asked to maximize (1 - x^2)^(1/2) * (1 + x):
max{sqrt(1 - x^2) (1 + x)} = (3 sqrt(3))/4 at x = 1/2
The area of the ellipse is pi * (3 sqrt(3))/4
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Posted by Charlie
on 2024-02-21 08:10:04 |
solution
