Mathematics, 21.03.2020 20:39 dezmondpowell

A Norman window is a window with a semi-circle on top of regular rectangular window. What should be the dimensions of the window to allow in as much light as possible, if there are only 12 ft of the frame material available?

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Using the perimeter and area formular for a rectangle and semicircle, the dimension of the window should have a length of 1.68 ft and the width should also be 1.68 ft

Using the Parameters :

Side lengths of the window = LRadius of semicircle = rwidth = 2r

Perimeter of semicircle = πr + 2r

Hence, perimeter of window can be expressed thus :

2L + 2r + πr = 12 - - - - (1)

Area of window :

2rL + 0.5πr^2 - - - - - (2)

Making 2L the subject of the equation :

2L = 12 - 2r - πr - - - (3)

Substitute 2L into (2)

A = r(12 - 2r - πr) + 0.5πr^2

A = 12r - 2r^2 - πr^2 + 0.5πr^2

A = 12r - 2r^2 - 0.5πr^2

Take the derivative of A to obtain maximum value of r

dA/dr = 12 - 4r - πr = 0

4r + πr = 12

r = 12 / ( 4 + π)

r = 12 / (4 + 3.14)

r = 1.68 ft.

Recall, width = 2r

width of the window = 2 * 1.68 = 3.36 ft.

From (3) :

2L = 12 - 2r - πr

12 - 2*1.68 - 1.68(3.14)

2L = 3.3648

L = 3.3648 / 2

L = 1.68.

Hence, the dimension of the window should be : 1.68 by 1.68

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Answer from: ltawiah8393

Side lengths = 1.68 ft and width = 3.36 ft.

Step-by-step explanation:

Let the side lengths of the window be L and the width = 2r ( r is also the radius of the semi-circle).

So we have

Perimeter = 2L + 2r + πr = 12

Area = 2rL + 0.5πr^2

From the first equation

2L = 12 - 2r - πr

Substitute for 2L in the equation for the area:

A = r(12 - 2r - πr) + 0.5πr^2

A = 12r - 2r^2 - πr^2 + 0.5πr^2

A = 12r - 2r^2 - 0.5πr^2

We need to find r for the maximum area:

Finding the derivative and equating to zero:

A' = 12 - 4r - πr = 0=

4r + πr = 12

r = 12 / ( 4 + π)

r = 1.68 ft.

So the width of the window = 2 * 1.68 = 3.36 ft.

Now 2L = 12 - 2r - πr

= 12 - 2*1.68 - 1.68π

= 3.36

L = 1.68.

Answer from: Quest

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Which statements correctly describe the graph of the function f(x)=x^3-2x^2-19x+20? check all that

Answer from: Quest

no association

step-by-step explanation:

if you have eyes and you can use them efficiently you can see that it just looks like random dots like do a line of best fit and like they're so spread out it's such a weak association you might as well consider it nonexistent

Which statement best describes the association between variable x and variable y? perfect negative

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A Norman window is a window with a semi-circle on top of regular rectangular window. What should be...