SOLUTION: A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See the figure below.) If the

Algebra.Com
Question 1116692: A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See the figure below.) If the perimeter of the window is 32 ft, find the value of x so that the greatest possible amount of light is admitted.
Found 3 solutions by josgarithmetic, greenestamps, ikleyn:
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
, to account for the round part of the perimeter.



If the other rectangular dimension is y, then accounting for perimeter 32,


Solving that for y:




Area





.
to continue, finding and setting derivative to 0 would be the best way...
.
.
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


I can't see your figure; so I guessed that x is the width of the rectangle (and therefore the diameter of the semicircle). And I used y for the height of the rectangle.

We want to find the value of x that makes the greatest possible amount of light admitted through the window, given that the perimeter of the window is 32 feet. That means we want to maximize the area of the window.

So we need an expression in terms of a single variable (preferably x) for the area of the window.

The perimeter of the window, 32 feet, is the width of the rectangle, plus twice the height of the rectangle, plus the circumference of the semicircle:



The area of the window is

We can solve the equation for the perimeter for y in terms of x and substitute into the formula for the area to get the area in terms of x only.




Then the area of the window in terms of x only is




We differentiate and set the derivative equal to zero to find the value of x that maximizes the area.




= 8.96 ft, to 2 decimal places
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Let x = width and diameter;

    y = height of the rectangle part.


Then the perimeter   

P =   ====>   +  = 32  ====>  y = .


The area A =  +  =  +  =  -  -  +  =  +  - 



Then  the condition for the maximum area   = 0  takes the form


 = 0,   or    = 16  ====> x =  =  = 8.96 ft.


Answer.  The maximum area is at x = 8.96 ft.

Solved.

==============

Be aware:   The method on how  @josgaritmetic  is doing it   I S   I N C O R R E C T.



RELATED QUESTIONS

A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the... (answered by ankor@dixie-net.com)
A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter (answered by Theo)
A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter (answered by ankor@dixie-net.com)
A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter (answered by greenestamps)
A norman window has the shape of a rectangle surmounted by a semicircle of a diameter... (answered by solver91311)
A Norman window has the shape of a rectangle surmounted by a semicircle of diameter equal (answered by htmentor)
A Norman window has the shape of a rectangle surmounted by a semicircle, as shown in the... (answered by josgarithmetic,ikleyn)
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of... (answered by ankor@dixie-net.com)
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of... (answered by ikleyn)