A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 40 feet?

5.60 ft and 5.60 ft

To find the largest possible area of the Norman window, we need to express the area in terms of a single variable so that we can differentiate it and find its maximum value.

Let's assume the width of the rectangle is w, then the diameter of the semicircle is also w. The radius of the semicircle is half the diameter, so it would be w/2.

The perimeter of the Norman window is equal to the sum of the perimeter of the rectangle and half the circumference of the semicircle.

Perimeter = (2 * width) + (π * radius)

Given that the perimeter is 40 feet, we can write the equation as:

40 = 2w + (π * (w/2))

Let's solve this equation to find the value of w:

40 = 2w + (π * w/2)
40 = 2w + (πw/4)

Multiply both sides of the equation by 4 to remove the fraction:

160 = 8w + πw

Now, factor out w:

160 = (8 + π)w

Divide both sides of the equation by (8 + π):

w = 160 / (8 + π)

Now that we have the width, we can calculate the area of the Norman window.

Area = (Area of the rectangle) + (Area of the semicircle)

Area = width * (w/2) + (π * (w/2)^2 / 2) [area of rectangle + area of semicircle]

Substituting the value of w, we get:

Area = (160 / (8 + π)) * ((160 / (8 + π))/2) + (π * ((160 / (8 + π))/2)^2 / 2)

Simplify this expression to get the final answer.

To find the area of the largest possible Norman window, we first need to determine the dimensions of the window. Let's denote the width of the rectangle as "w" and the radius of the semicircle as "r."

Based on the given information, we know that the diameter of the semicircle is equal to the width of the rectangle, so the radius is half the width, r = w/2.

Now, let's calculate the perimeter of the Norman window. The perimeter is the sum of the lengths of all sides.

The perimeter, P, can be calculated as follows:
P = 2w + πr + r,

where 2w represents the two horizontal sides of the rectangle, πr represents the curved part of the semicircle, and r represents the vertical side of the semicircle.

Given that the perimeter is 40 feet, we have the following equation:
40 = 2w + πr + r.

Now, we can substitute r = w/2 into the equation:
40 = 2w + π(w/2) + (w/2).

Simplifying the equation:
40 = 2w + (π + 1)w/2.

Combining like terms:
40 = (4 + π)w/2.

Multiply both sides of the equation by 2/(4 + π):
40 * 2 / (4 + π) = w.

Substituting the numerical value of π (approximately 3.14):
40 * 2 / (4 + 3.14) = w,
80/(7.14) = w,
w ≈ 11.19.

Now, since the radius is half the width, r = w/2:
r ≈ 11.19/2,
r ≈ 5.59.

Finally, we can calculate the area of the largest possible Norman window. The area is the sum of the area of the rectangle and half the area of the semicircle:

Area = w * r + (π * r^2)/2,
Area = 11.19 * 5.59 + (3.14 * 5.59^2)/2,
Area ≈ 124.16 + 49.33,
Area ≈ 173.49 square feet.

Therefore, the area of the largest possible Norman window with a perimeter of 40 feet is approximately 173.49 square feet.

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