To solve this problem, we need to determine the dimensions of the window ($w$ and $h$) that minimize the perimeter, given that the cross-sectional area is $A$. Here's a step-by-step outline:

Problem Breakdown:

1. Shape Details:

   • The window consists of a rectangular part of width $w$ and height $h$.
   • On top of the rectangle is an isosceles triangle with height $T = 0.4w$.

2. Constraints:

   • The total cross-sectional area $A$ is given by: $A = w \cdot h + \frac{1}{2} \cdot w \cdot T$ Substituting $T = 0.4w$, the area becomes: $A = w \cdot h + \frac{1}{2} \cdot w \cdot (0.4w)$ $A = w \cdot h + 0.2w^2$

3. Objective:

   • Minimize the perimeter of the window:
     • Perimeter includes the sides of the rectangle and the two equal sides of the triangle: $P = 2h + w + 2s$ where $s$ is the slant height of the triangle. Using the Pythagorean theorem: $s = \sqrt{\left(\frac{w}{2}\right)^2 + T^2} = \sqrt{\left(\frac{w}{2}\right)^2 + (0.4w)^2}$ $s = \sqrt{0.25w^2 + 0.16w^2} = \sqrt{0.41w^2} = 0.64w$ Therefore: $P = 2h + w + 2(0.64w) = 2h + 2.28w$

4. Minimization:

   • Use the area constraint to express $h$ in terms of $w$: $h = \frac{A - 0.2w^2}{w}$
   • Substitute $h$ into the perimeter formula: $P = 2\left(\frac{A - 0.2w^2}{w}\right) + 2.28w$ Simplify: $P = \frac{2(A - 0.2w^2)}{w} + 2.28w$ $P = \frac{2A}{w} - 0.4w + 2.28w$ $P = \frac{2A}{w} + 1.88w$

5. Optimization:

   • Minimize $P$ by differentiating with respect to $w$ and setting $\frac{dP}{dw} = 0$: $\frac{dP}{dw} = -\frac{2A}{w^2} + 1.88$ $-\frac{2A}{w^2} + 1.88 = 0$ $\frac{2A}{w^2} = 1.88$ $w^2 = \frac{2A}{1.88}$ $w = \sqrt{\frac{2A}{1.88}}$

6. Determine $h$:

   • Substitute $w$ back into the equation for $h$: $h = \frac{A - 0.2w^2}{w}$

Would you like further clarification or numerical calculations?

Here are 5 related questions for practice:

1. How does the height of the triangle affect the overall perimeter?
2. Why do we use calculus to minimize the perimeter in this case?
3. What happens to the dimensions if the area $A$ increases?
4. Can the method work if the triangle’s height were a different fraction of $w$?
5. How can we verify that our solution gives a minimum and not a maximum for $P$?

Tip: Always check the physical feasibility of your dimensions after solving optimization problems!