Consider a window the shape of which is a rectangle of height h surmounted a triangle having a height T that is 1.3 times the width w of the rectangle (as shown in the figure below).
(pictures of a pentagon)
If the cross-sectional area is A, determine the dimensions of the window which minimize the perimeter.
h=_____
w=_____
To determine the dimensions of the window that minimize the perimeter, we need to first understand the relationship between the variables and find an expression for the perimeter in terms of those variables.
Let's break down the shape into its components: a rectangle and a triangle.
1. Rectangle:
The height of the rectangle is given as h.
2. Triangle:
The height of the triangle is given as T, and it is mentioned that the height T is 1.3 times the width w of the rectangle. Therefore, we can express the width of the rectangle as w = T / 1.3.
To find the perimeter, we need to consider the sum of all the sides of the rectangle and the triangle.
Perimeter = 2*(length + width of rectangle) + sum of the three sides of the triangle
Since one side of the triangle is given as the width of the rectangle, we only need to consider the other two sides of the triangle.
Now, let's calculate the length of the triangle using the Pythagorean theorem:
(Length of triangle)^2 = (Height of triangle)^2 + (Width of rectangle)^2
Plugging in the given values, we have:
(Length of triangle)^2 = T^2 + (T/1.3)^2
Now, we can express the perimeter in terms of the variables h and T:
Perimeter = 2*(h + T/1.3) + Length of the triangle + Width of the rectangle
Next, we need to differentiate the perimeter expression with respect to T and set it equal to zero to find the minimum perimeter.
After calculating the derivative and solving for T, you can substitute the value of T into the expression for w to find the width.
Finally, you can substitute the found values of h and w into the perimeter expression to find the minimum perimeter.
Unfortunately, without the specific measurements for h and T, we cannot provide the exact values for h and w that minimize the perimeter.