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Math Help - Optimization Problem

  1. #1
    a.a
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    Optimization Problem

    A rectangle PQRS is inscribed in the region between the x-axis and the part of the graph y= cos 4x specified by -pi/8 < x< pi/8. ( the < sign is less than or equal to). Determine the coordinates of P for which the perimeter of PQRS is a maximum.

    Im not even sure where to begin...
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    Quote Originally Posted by a.a View Post
    A rectangle PQRS is inscribed in the region between the x-axis and the part of the graph y= cos 4x specified by -pi/8 < x< pi/8. ( the < sign is less than or equal to). Determine the coordinates of P for which the perimeter of PQRS is a maximum.

    Im not even sure where to begin...
    Did you draw a diagram?

    the height of the rectangle is cos(4x), the length of the rectangle is 2x (we denote the distance from the y-axis to the right bottom corner of the rectangle to be x)

    now that you know the height and length of the rectangle, can you continue?
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Jhevon View Post
    Did you draw a diagram?

    the height of the rectangle is cos(4x), the length of the rectangle is 2x (we denote the distance from the y-axis to the right bottom corner of the rectangle to be x)

    now that you know the height and length of the rectangle, can you continue?
    P=2x+2\cos(4x)...assuming you use the same distance x from both sides which it looks like you do...you can do this considering the eveness of cos(x)
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    ...you can do this considering the eveness of cos(x)
    yes, indeed
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  5. #5
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    Quote Originally Posted by Mathstud28 View Post
    P=2x+2\cos(4x)...assuming you use the same distance x from both sides which it looks like you do...you can do this considering the eveness of cos(x)

    wouldnt P = 2 cos 4x + 4x? because 2 x is the length and P is 2l + 2w??
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by a.a View Post
    wouldnt P = 2 cos 4x + 4x? because 2 x is the length and P is 2l + 2w??
    yes
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  7. #7
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    something is wrong...

    ok so P= 2cos 4x + 4x
    then P' = -2 sin 4x + 4
    so at max. -2 six 4x + 4 = 0
    sin 4x = 2.... where do i go from here?
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by a.a View Post
    ok so P= 2cos 4x + 4x
    then P' = -2 sin 4x + 4
    so at max. -2 six 4x + 4 = 0
    sin 4x = 2.... where do i go from here?
    Do what you usually do..solve for 0...test that value in second derivative...if you forget this stuff look at this page...it has a huge amount of useful calculus review information

    http://www.mathhelpforum.com/math-he...-tutorial.html
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by a.a View Post
    ok so P= 2cos 4x + 4x
    then P' = -2 sin 4x + 4
    so at max. -2 six 4x + 4 = 0
    sin 4x = 2.... where do i go from here?
    by the way, your derivative is wrong, you need the chain rule to derive cos(4x)

    of course what you have has no solutions (sin(4x) never attains 2, its maximum is 1), you must have realized something was off
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  10. #10
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by a.a View Post
    do you mean solve for x... how are we supposed to do that? sin inverse of 2 is undefined
    P=2cos(4x)+4x\Rightarrow{P'=-8sin(4x)+4} setting that equal to zero we have x=\frac{arsin\bigg(\frac{1}{2}\bigg)}{4}+\frac{\pi  }{2}n,n\in\mathbb{Z}...could someone second me on that?
    Last edited by Mathstud28; April 20th 2008 at 07:19 PM.
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  11. #11
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    -8 sin 4x +4 = 0
    sin 4x = 1/2
    4x = pi/6
    x= pi/24
    ...........................
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  12. #12
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by a.a View Post
    -8 sin 4x +4 = 0
    sin 4x = 1/2
    4x = pi/6
    x= pi/24
    ...........................
    ................................Yeah.............. .....................I didnt divide 4/8 wrong .........so the answer is \frac{\pi}{24}+\frac{\pi}{2}n,n\in\mathbb{Z}
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  13. #13
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by a.a View Post
    -8 sin 4x +4 = 0
    sin 4x = 1/2
    4x = pi/6
    x= pi/24
    ...........................
    ok...and now that you have that evaluate f''\bigg(\frac{\pi}{24}\bigg)...if its negative you have a max...if its positive you have a min
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  14. #14
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    yip... P'' is -ve hence P is concave down and its a max, so for the given domain the value for x would be pi/24..
    thanks for all the help... sorri for all the problems with this one...
    i need help with another one im gonna post... if you can please take a look
    thanks agian
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  15. #15
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by a.a View Post
    yip... P'' is -ve hence P is concave down and its a max, so for the given domain the value for x would be pi/24..
    thanks for all the help... sorri for all the problems with this one...
    i need help with another one im gonna post... if you can please take a look
    thanks agian
    Anytime!
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