1. ## Optimization Problem...

Consider a window the shape of which is a rectangle of height h surmounted a triangle having a height T that is 1.5 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 minimize the perimeter.

h=?
w=?

2. We can let $T=\frac{3}{2}w; \;\ \frac{2}{3}T=w$

$A_{rectangle}=h(\frac{2}{3}T)$

$A_{triangle}=\frac{1}{3}T^{2}$

$Total \;\ area=h(\frac{2}{3}T)+\frac{1}{3}T^{2}$...[1]

Therefore, the perimeter can be expressed as

$P=2h+\frac{2T}{3}+2\sqrt{(\frac{T}{3})^{2}+T^{2}}$...[2]

Continue by expressing P in terms of T alone. Then, differentiate, set to 0 and solve for T. You can also express it in terms of w instead of T.

thanks!