Probem21Hannah+W.,+Ian



21. Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.
 * the rectangle of largest area that can be inscribed in a circle will be a square
 * if a square is inscribed in a circle then
 * let s=side length
 * s^2 = 2r^2
 * s=r sqrt(2)
 * 

Answer: the rectangle would be a square with sides r sqrt(2)

What is the objective function? Are you trying to maximize it or minimize it? What other restrictions are you given? YOu need to consider this as a problem in which you are looking for absoltue extrema (and you know the drill)

Ms. Sweeney, I don't understand any other practical way to do this. Am I supposed to assume that I don't know it will be a square? A square with sides r sqrt(2) is the only square that would inscribe into a circle. Ian R. Wilson:

21. Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.  It is thusly given that

A=ab, and that a^2+b^2=c^2, and that, since c describes the diameter of the circle as well, a^2+b^2=4r^2.

However we know that A=ab and thus it follows that A^2=a^2b^2. And since it could not be more obvious that the highest real positive value of A (we are restricted to real positive values as these objects exist in euclidean space) will yield the highest value A^2, the search remains the same, we shall thus refer to A^2 as f(x) and carry on.

a^2=4r^2 -b^2

and b^2=4r^2-a^2

therefore:

f(x)=( 4r^2 -b^2) b^2

f(x)=4r^2  b^2 - b^4 f'(x)=8r^2b - 4b^3 0= 8r^2b-4b^3 <span style="color: rgb(0, 0, 0);">8r^2b= <span style="color: rgb(0, 0, 0);">4b^3 2r^2=b^2 b=sqrt(2)r

and therefore, since a^2+b^2=4r^2

2r^2+a^2=4r^2 a^2=2r^2 therefore a=b and therefore the largest rectangle that may be inscribed in a circle with radius r, is a sqare with sides sqrt(2)r.

(ms. sweeney): you only need to consider the outer reaches of b: what is the smallest value of b? the largest? how does your objective function behave at these values? <span style="color: rgb(0, 0, 0);"> <span style="color: rgb(0, 0, 0);">