Consider a window the shape of which is a rectangle...

Question

Consider a window the shape of which is a rectangle of height hh surmounted by a triangle having a height T that is 1.1 times the width w of the rectangle (as shown in the figure below). 

If the cross-sectional area is, A, determine the dimensions of the window which maximize the perimeter.

h = ___

w = ___

Answer 1

assuming A=h*w

by given, sides of triangle are w*(√5)/2

Perimeter P=w*(√5+1)+2A/w

find P' and set to 0

have w²=2A/(√5+1)

w is positive,,,w is positive square root of 2A/(√5+1)for critical point

note: P'' positive when w is positive so critical point is for minimum perimeter

maximum perimeter is at end points,,,,w=0 or h=0 both of which are not feasible for A=h*w