A Norman window consists of a rectangle capped by a semi-circular regional. The perimeter of some particular Norman window must be 30 meters. The radius of the semi-circular region is "x" meters. The height of the rectangle is "h" meters. Find the values pf "x" and "h" such that this Norman window has a maximum area. That is, the Norman window will permit the most light to shine through.

If the radius is x, then the base of the rectangle is 2x

let the height of the rectangle be h

Perimeter = 30 = (1/2)(2πx) + 2h + 2x
30 = πx + 2h + 2x
h = (30 - πx - 2x)/2

Area = A = 2xh + (1/2)πx^2
= 2x(30 - πx - 2x)/2 + (1/2)πx^2

I will let you finish it ...
simplify a bit, then differentiate,
set the derivative equal to zero and solve for x

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To find the values of "x" and "h" that maximize the area of the Norman window, we need to express the area of the window in terms of "x" and "h" and then apply calculus techniques to find the maximum.

Let's begin by breaking down the Norman window into its rectangular and semi-circular regions:

1. The rectangular region has a height of "h" and a width equal to the total width of the window minus the diameter of the semi-circle (2x). Therefore, the width of the rectangle is (2x - 2r), where "r" represents the radius of the semi-circle.

2. The semi-circular region has a radius of "x".

Now, let's calculate the area of each region:

1. Area of the rectangular region:
- A_rectangular = width * height
- A_rectangular = (2x - 2r) * h

2. Area of the semi-circular region:
- A_semi-circular = π * radius^2 / 2
- A_semi-circular = π * x^2 / 2

To find the total area of the Norman window, we sum the areas of the rectangular and semi-circular regions:

A_total = A_rectangular + A_semi-circular
= (2x - 2r) * h + π * x^2 / 2

Now, we need to maximize A_total with respect to both "x" and "h". To do this, we take partial derivatives with respect to both variables and set them equal to zero to find critical points:

∂A/∂x = 0 and ∂A/∂h = 0

Differentiating A_total with respect to "x" and setting it to zero:

∂A_total/∂x = 2h - 2r + πx = 0 ............ (1)

Differentiating A_total with respect to "h" and setting it to zero:

∂A_total/∂h = 2x - 2r = 0 ............ (2)

From equation (2), we can see that 2x - 2r = 0, which implies 2x = 2r or x = r.

Now, substituting this value of "x" into equation (1):

2h - 2r + πr = 0
2h = πr
h = πr / 2

Therefore, to maximize the area of the Norman window, "x" should be equal to "r" and "h" should be equal to πr / 2.

In conclusion, for the Norman window to have the maximum area, the radius of the semi-circular region ("x") and the height of the rectangular region ("h") should be equal to each other and are given by:

x = r
h = πr / 2