3.4 App. of quadratic functions (AM-GM)

Consider a rectangle with area 20. If the minimum perimeter of the rectangle is can be written as [math]a\sqrt{b}[/math] where [math]a[/math] and [math]b[/math] are integers and [math]b[/math] is a prime number, what is the value of [math]ab[/math]?

Let [math]x[/math] and [math]y[/math] be positive numbers satisfying [math]xy=25[/math], what is the minimum of [math]4x+9y+5[/math]?

Consider a line [math]\ell: 5x+3y=4[/math]. If point [math]P(a,b)[/math] is on [math]\ell[/math] where [math]a, b>0[/math], what is the maximum of [math]ab[/math]?

Let [math]A(0,0)[/math] and [math]B(5,0)[/math] be two points on the [math]xy[/math]-plane. Point [math]P[/math] is on the curve [math]y=\sqrt{r^2-x^2}[/math] such that [math]\overline{PA}[/math] is perpendicular to [math]\overline{PB}[/math] where [math]0<r<5[/math].

(a) If the area of triangle [math]APB[/math] is [math]\dfrac{1}{2}\sqrt{f(r)}[/math], what is [math]f(r)[/math]?

(b) Use the arithmetic-geometric mean inequality to find the maximum area of triangle [math]APB[/math].

Consider a right-angle triangle with length [math]a, b, c[/math] where [math]c[/math] is the length of the hypotenuse. If  [math]a+b\leq\sqrt{k}c[/math] where [math]k[/math] is an integer, what is the value of [math]k[/math]?