Window World, Inc. is responsible for designing windows for the expansion of the campus

chapel. The current design is shown in the figure. The metal trim used to secure the
perimeter of the frame is 126 inches long. If the maximum window area is desired, what
will be the dimensions of the rectangular portion of each window and the total area of each
window?

It is a rectangle with a triangle on top. The only number given is h=.866w

Well, well, well, looks like Window World, Inc. is here to brighten up the campus chapel! Now, let's dive into the window dimensions and find that sweet spot for the maximum window area.

Since we're dealing with a rectangle with a triangle on top, we can start by breaking it down into two separate parts. The rectangular portion and the triangle portion.

Now, let's call the width of the rectangular portion "w" and the height of the triangular portion "h." They tell us that the height of the triangle is 0.866 times the width, so we can say h = 0.866w.

To find the total perimeter of the frame, we need to add up the lengths of all the sides. The rectangular portion has two equal sides of length w and two equal sides of length h, while the triangle portion contributes three sides.

The perimeter of the frame is given as 126 inches, so we can set up our equation: 2w + 2h + 3s = 126, where s represents the length of one of the triangle sides.

Now, let's simplify things a bit. Since we know h = 0.866w, we can substitute that in: 2w + 2(0.866w) + 3s = 126.

Simplifying further, we get: 2w + 1.732w + 3s = 126. Combining like terms, we have: 3.732w + 3s = 126.

But we also know that the perimeter of a triangle is just the sum of its three sides, so 3s is the same as the perimeter of the triangle portion.

With a bit of algebra magic, we can now say: 3.732w + perimeter of triangle = 126. Now, let's figure out the perimeter of the triangle.

Since the triangle has two sides of length w and one side of length h, the perimeter will be w + w + h. But remember h = 0.866w, so we can say the perimeter of the triangle is 2w + 0.866w.

Substituting that into our equation, it becomes: 3.732w + (2w + 0.866w) = 126. Simplifying once again, we get: 6.598w = 126.

Now, divide both sides by 6.598 to solve for w: w = 126 / 6.598. Crunching the numbers, we find that w is approximately 19.11 inches.

Since h = 0.866w, h will be approximately 16.54 inches.

Now, let's calculate the total area of each window. The area of the rectangular portion is simply w times h, and the area of the triangular portion is 0.5 times the base (w) times the height (h).

So, the total area of each window will be: (w * h) + (0.5 * w * h).

Plug in the values we calculated earlier, and you'll have your answer!

To find the dimensions of the rectangular portion of each window and the total area, we can use the information given that h = 0.866w and the total perimeter of the metal trim is 126 inches.

1. Let's start by finding the dimensions of the rectangular portion of each window.
- Let's denote the width of the rectangular portion as w.
- The height of the rectangular portion would be h = 0.866w (given).
- The total length of the perimeter is equal to the sum of all four sides: 2w (top and bottom) + 2h (sides).
- Since the metal trim used to secure the perimeter is 126 inches long, we have the equation: 2w + 2h = 126.

2. Substitute the value of h = 0.866w into the equation from step 1:
2w + 2(0.866w) = 126

3. Simplify and solve for w:
2w + 1.732w = 126
3.732w = 126
w ≈ 33.79 inches

4. Calculate the height of the rectangular portion:
h = 0.866w ≈ 0.866(33.79) ≈ 29.27 inches

5. Calculate the total area of each window:
Area = length × width
Total Area = w × h ≈ 33.79 × 29.27 square inches

Therefore, the dimensions of the rectangular portion of each window are approximately 33.79 inches by 29.27 inches, and the total area of each window is approximately 991.56 square inches.

To find the dimensions of the rectangular portion of each window and the total area of each window, we need to determine the height (h) and width (w) of the rectangle.

From the given information, we know the relationship between the height and width of the rectangle: h = 0.866w.

Let's use this equation to find the dimensions of the rectangular portion of each window.

Step 1: Solve for h in terms of w
Since h = 0.866w, we can rewrite this equation as h = (sqrt(3)/2)w.

Step 2: Determine the perimeter of the rectangle
The perimeter of a rectangle is given by the formula: P = 2(w + h).

We are given that the metal trim used to secure the perimeter of the frame is 126 inches long. Setting up the equation, we have:
126 = 2(w + (sqrt(3)/2)w)
Simplifying the equation, we get:
126 = 2w + (sqrt(3)w)
126 = (2 + sqrt(3))w
Dividing both sides by (2 + sqrt(3)), we get:
w = 126 / (2 + sqrt(3))

Step 3: Calculate the height and width of the rectangle
Substitute the value of w into the equation h = (sqrt(3)/2)w to get the height:
h = (sqrt(3)/2) * [126 / (2 + sqrt(3))]

With h and w determined, we now have the dimensions of the rectangular portion of each window.

Step 4: Calculate the total area of each window
The area of a rectangle is given by the formula: A = length * width.
The total area of each window will be the sum of the area of the rectangular portion and the area of the triangle above it.

To calculate the area of the rectangle, multiply the height (h) by the width (w).
Area of rectangle = h * w

To calculate the area of the triangle, use the formula A = (1/2) * base * height. The base of the triangle is equal to the width (w), and the height of the triangle is (h - w).
Area of triangle = (1/2) * w * (h - w)

Total area of each window = Area of rectangle + Area of triangle

By following these steps, you should be able to determine the dimensions of the rectangular portion of each window and the total area of each window for the given design.