Question 750024
Both curves and the rectangle QRST are symmetrical with respect to the y-axis.
TI'll call the coordinates of P (a,0), to distinguish that x-coordinate value {{{a}}} from the variable {{{x}}.
he x-coordinates of points Q, P and T are the same {{{x=a}}}.
The x-coordinates of points R and S are the same {{{x=-a}}}.
The y-coordinate of points Q and R, on curve {{{y=-x^2+4}}} is {{{y=-a^2+4}}}.
The y-coordinate of points S and T, on curve {{{y=x^2-4}}} is {{{y=a^2-4}}}.
The width ST (or QR) of the rectangle is {{{w=2a}}}.
The height RS (or QT) of the rectangle is {{{h=-a^2+4-(a^2-4)=-a^2+4-a^2+4=-2a^2+8=2(-a^2+4)}}}.
The perimeter of the rectangle is
{{{alpha=2(w+h)=2(2a+2(-a^2+4))=-4a^2+4a+16}}}
{{{alpha}}} is a quadratic function in {{{a}}}
It's maximum is at {{{a=-4/(2(-4))=highlight(1/2)}}}
because a parabola such as {{{y=ax^2+bx+c}}} has a vertex as {{{x=-b/2a}}}
The equation in vertex form would be
{{{alpha=-4(a-1/2)^2+highlight(17)}}}
{{{-4(a-1/2)^2+17=-4(a^2-a+1/4)+17=-4a^2+4a-1+17=-4a^2+4a+16}}}
So the maximum value of {{{alpha}}} happens at {{{highlight(A=1/2)}}}
and the maximum value of {{{alpha}}} is {{{highlight(B=17)}}}