You have a fixed amount of window trim for a particular window. The window is a rectangle with two circles. All circles are the same size and have a diameter equal to the width of the window. (Not...

2. Speculate about what the result would be if there were n such circles. If you would like to prove your speculation to be correct (or not!), do so algebraically on a SEPARATE page that also has your name.

Extracted text: You have a fixed amount of window trim for a particular window. The window is a rectangle with two circles. All circles are the same size and have a diameter equal to the width of the window. (Not necessarily the same as the height!) Let P represent the fixed perimeter. Let r represent the radius of the circle. Let y represent the height of the rectangle. Let A represent the Area of the combined figure. (a) Write a constraint involving the perimeter; that is, write P in terms of r and y. (b) Write an equation for the Area as a function of r and y. (c) Solve the constraint for y (or for 2y) and substitute into the Area equation. (d) Take the derivative and set it equal to zero. Solve for P. (e) Substitute for P in the equation in part (c) and solve for y. (f) What is the RATIO of the height y to width 2r for the rectangle that maximizes the area of the figure? (g) What is the interpretation? Why does it make sense? (h) Does it matter WHERE the two circles are? Do they need to be attached to the rectangle? (Use SENTENCES!)