Maxima/Minima
Problem 3
An ellipse having perimeter of 2015π is to be draw on a paper whose sides are 2020π x 2020π. If you want to land the tip of your pen exactly to the perimeter of the ellipse, find the maximum probability of doing this.
Express answer in fraction form.
To find the maximum probability of landing the tip of your pen exactly on the perimeter of the ellipse, we can use the concept of calculus and optimization.
Let's denote the semi-major axis of the ellipse as a and the semi-minor axis as b. The perimeter of an ellipse is given by the formula:
P = 2π √( (a^2 + b^2) / 2 )
In this case, we are given that the perimeter of the ellipse is 2015π. Therefore, we can write the equation:
2015π = 2π √( (a^2 + b^2) / 2 )
Simplifying the equation, we get:
√( (a^2 + b^2) / 2 ) = 2015 / 2
Squaring both sides of the equation to eliminate the square root, we have:
(a^2 + b^2) / 2 = (2015 / 2)^2
Simplifying further:
a^2 + b^2 = (2015 / 2)^2 * 2
a^2 + b^2 = 2015^2
Now, we need to maximize the probability of landing the tip of the pen on the perimeter of the ellipse. The probability is maximized when the ellipse has the maximum area, as the area represents all possible locations for the tip of the pen.
The area of an ellipse is given by the formula:
A = πab
Our goal is to maximize A, so we can rewrite this equation as follows:
A = πa √( a^2 - (2015^2 - a^2) ) / a
Simplifying further:
A = πa √( 2a^2 - 2015^2 ) / a
The probability of landing the tip of the pen on the perimeter of the ellipse is given by the ratio of the perimeter of the ellipse to the area:
P = (perimeter of the ellipse) / (area of the ellipse)
Substituting the formulas for the perimeter and area of the ellipse, we have:
P = (2π √( (a^2 + b^2) / 2 )) / (πa √( 2a^2 - 2015^2 ) / a)
Simplifying and canceling common terms, we get:
P = 2 / √( 2a^2 - 2015^2 )
To maximize P, we need to find the maximum value of √( 2a^2 - 2015^2 ).
To do this, we take the derivative of √( 2a^2 - 2015^2 ) with respect to a and set it equal to zero:
(d/dx) √( 2a^2 - 2015^2 ) = 0
Simplifying further, we have:
a = ± 2015 / √2
Since the sides of the paper are 2020π x 2020π, a must be less than or equal to half of the length of the sides. Therefore, we take the positive value of a, which is:
a = 2015 / √2
Finally, substituting this value back into the expression for P, we have:
P = 2 / √( 2 * (2015 / √2)^2 - 2015^2 )
Simplifying the expression for P, we get:
P = 2 / √( 4030^2 - 2015^2 )
P = 2 / √( 4030^2 / 2 )
P = 2 / ( 4030 / √2 )
Simplifying further, the maximum probability of landing the tip of your pen exactly on the perimeter of the ellipse is:
P = 2√2 / 4030
Therefore, the answer is 2√2/4030 in fraction form.