Consider a window the shape of which is a rectangle of height h surmounted by a triangle having a height T that is 0.5 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.
To find the dimensions of the window that minimize the perimeter, we need to express the perimeter in terms of one variable and then differentiate it to find the minimum value.
Let's break down the problem step by step:
Step 1: Define the variables.
- Let h be the height of the rectangle.
- Let w be the width of the rectangle.
- Let T be the height of the triangle.
- Let P be the perimeter of the window.
- Let A be the cross-sectional area of the window.
Step 2: Determine the expressions for the dimensions.
- The width of the triangle (base) is equal to w.
- The area of the rectangle is given by A_rectangle = h * w.
- The area of the triangle is given by A_triangle = (1/2) * w * T.
Since the cross-sectional area A is given, we can express h in terms of A and w:
A = A_rectangle + A_triangle
Substituting the expressions for the areas, we get:
A = h * w + (1/2) * w * T
Now, we can solve this equation for h:
h = (A - (1/2) * w * T) / w
Step 3: Express the perimeter in terms of one variable.
The perimeter P is given by:
P = 2w + 2h + T
Substituting the expression for h from Step 2, we get:
P = 2w + 2[(A - (1/2) * w * T) / w] + T
Simplifying further:
P = 2w + 2A/w - T + T
P = 2w + 2A/w
Step 4: Differentiate the expression for P with respect to w and set it equal to zero to find the minimum.
dP/dw = 2 - 2A/w^2 = 0
Solving the above equation:
2 = 2A/w^2
w^2 = A
w = sqrt(A)
So, the width of the window that minimizes the perimeter is w = sqrt(A).
Step 5: Find the corresponding values of h and T.
Substitute the value of w = sqrt(A) back into the expression for h:
h = (A - (1/2) * sqrt(A) * T) / sqrt(A)
Simplify:
h = A / sqrt(A) - (1/2) * T
For the height of the triangle, we are given that T = 0.5w. Substituting w = sqrt(A):
T = 0.5 * sqrt(A)
So, the dimensions of the window that minimize the perimeter are:
- Width: w = sqrt(A)
- Height of the rectangle: h = A / sqrt(A) - 0.25 * sqrt(A)
- Height of the triangle: T = 0.5 * sqrt(A)
Note that this solution assumes that A > 0 and the dimensions of the window are physically feasible.