Question 1134762
given:
 The perimeter of the window is {{{12}}} feet. The diameter of a circle and the length of a rectangle are {{{x}}}  the width is {{{y}}}.

1) Write the area {{{A}}} of the window as a function of {{{x}}}.


The perimeter is {{{12}}} ft.

{{{P = 2y + x +1/2(2pi*r) }}}

{{{P= 2y + x + pi(x/2)}}}

{{{P= 2y +(1 +pi/2)x}}} .......given {{{P= 12}}}.

{{{12= 2y +(1 +pi/2)x}}}.....solve for {{{y}}}

{{{2y =12-(1 +pi/2)x}}}

{{{y =6-((2 +pi)x/2)/2}}}

{{{y =6-((2 +pi)x)/4}}}


We find an equation for maximum area.

{{{A = xy +(1/2)pi*r^2}}} ......{{{r=x/2}}}

{{{A = xy +(1/2)pi*(x/2)^2}}}
 
{{{A = xy +(1/2)pi*(x^2/4)}}}
 
{{{A = xy +(1/8)pi*x^2}}} 


Substitute in for {{{y}}}. Now

{{{A = x(6-((2 +pi)x)/4) +(1/8)pi*x^2 }}}

{{{A = 6x-(1/4)(2 +pi)x^2 +(1/8)pi*x^2 }}}

{{{A = -(pi*x^2)/8 - x^2/2 + 6x}}}

{{{A = -(1/8) (4 + pi) x^2+ 6x}}}->the area {{{A}}} of the window as a function of {{{x}}}


2) What dimensions will produce a window of maximum area?

Next compute {{{dA/dx}}}. Set it equal to zero and solve for {{{x}}}.

{{{dA/dx = 2(-1/8)(4 + pi)x+6}}}

{{{dA/dx = -(1/4) (4 + pi)x+6}}}

set ={{{ 0}}}.


{{{-(1/4) (4 + pi)x+6=0}}}

{{{(1/4) (4 + pi)x=6}}}

{{{ (4 + pi)x=6/(1/4)}}}

 {{{(4 + pi)x=24}}}

{{{x=24/(4 + pi)}}}
 
{{{x=24/(4 + 3.14) }}}

{{{x=24/7.14}}}

{{{x}}}≈ {{{3.35ft}}}

then {{{r=3.35ft/2=1.675}}}


Next find {{{y}}}.

{{{y =(6-(2 +pi)x)/4}}}

{{{y =(6-(2 +3.14)3.35)/4}}}

{{{y =(6-5.14*3.35)/4}}}

{{{y =6-(5.14*3.35)/4}}}

{{{y}}} ≈ {{{1.7ft}}}

dimensions that will produce a window of maximum area are:

{{{x}}}≈ {{{3.35ft}}}

{{{r}}}≈{{{1.675}}}

{{{y}}} ≈ {{{1.7ft}}}


check the perimeter first then see what would be maximum area

{{{12= 2y +(1 +pi/2)x}}}

{{{12= 2*1.7 +(1 +3.14/2)3.35}}}

{{{12= 2*1.7 +((2 +3.14)/2)3.35}}}

{{{12= 2*1.7 +(5.14*3.35)/2}}}

{{{12= 3.4 +8.6095}}}

{{{12= 12.0095}}}

{{{12= 12}}}-> true


now find maximum area

{{{A = xy +(1/8)pi*x^2 }}}

{{{A = 3.35*1.7 +(1/8)3.14*(3.35)^2}}}
 
{{{A = 5.695 +(1/8)3.14*11.2225}}}

{{{A = 5.695 +4.40483125}}}

{{{A = 5.695 +4.40483125}}}

{{{A =10.0998}}}

{{{A }}}≈ {{{10ft^2}}}