A "cross" is inscribed in a circle of radius 9 (see diagram below). The cross is symmetric, so each outer edge (the ones in purple) has the same length, say $\displaystyle {2}{x}$. We seek to find $\displaystyle {x}$ so that the area of the cross is maximized. You may move the slider to see the effect on the cross when $\displaystyle {x}$ is increased or decreased.

(a) Write the function $\displaystyle {A}{\left({x}\right)}$ that gives the area of the cross as a function of $\displaystyle {x}$. Let $\displaystyle {2}{x}$ be the length of an outer edge of the cross (the segments in purple).

      $\displaystyle {A}{\left({x}\right)}=$ Preview Question 6 Part 1 of 4  

(b) What is the open interval on which $\displaystyle {A}{\left({x}\right)}$ is defined? We'll insist that the interval is open so that we actually have a cross.

      Preview Question 6 Part 2 of 4  

(c) $\displaystyle {A}'{\left({x}\right)}=$ Preview Question 6 Part 3 of 4  

(d) What is the critical number for $\displaystyle {A}{\left({x}\right)}$ that lies in the interval identified in part (b)? Write an exact value, i.e. no decimal approximation.

      Preview Question 6 Part 4 of 4