Results 1 to 4 of 4

Math Help - integrals of trigonometric functions

  1. #1
    Junior Member
    Joined
    Jan 2009
    From
    vancouver
    Posts
    58

    integrals of trigonometric functions

    a thief tries to enter a building by placing a ladder over a 9-ft-high fence so it rests against the building,which is 2 ft back from the fence.

    what is the length f the shortest ladder and the ground? (let Q be the angle between the ladder and the ground) express the length of the ladder in terms of Q,

    and then find the value of Q that minimizes the length of the ladder.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    12,108
    Thanks
    970
    let x = distance from foot of the ladder to the fence

    f\cos{Q} = x+2

    \cot{Q} = \frac{x}{9}


    f\cos{Q} = 9\cot{Q} + 2

    f = 9\csc{Q} + 2\sec{Q}

    find \frac{df}{dQ} and minimize f
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jan 2009
    From
    vancouver
    Posts
    58
    after I find out the derivetive, how can I minimize the f? let that equals to 0,but I don't know how to solve the value for Q
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    12,108
    Thanks
    970
     <br />
f = 9\csc{Q} + 2\sec{Q}<br />

     <br />
\frac{df}{dQ} = -9\csc{Q}\cot{Q} + 2\sec{Q}\tan{Q} = 0<br />

    \frac{-9\cos{Q}}{\sin^2{Q}} + \frac{2\sin{Q}}{\cos^2{Q}} = 0

     <br />
\frac{-9\cos^3{Q}}{\sin^2{Q}\cos^2{Q}} + \frac{2\sin^3{Q}}{\sin^2{Q}\cos^2{Q}} = 0<br />

     <br />
-9\cos^3{Q} + 2\sin^3{Q} = 0<br />

     <br />
9\cos^3{Q} = 2\sin^3{Q} <br />

     <br />
\frac{9}{2} = \frac{sin^3{Q}}{\cos^3{Q}}<br />

    \frac{9}{2} = \tan^3{Q}

    \sqrt[3]{\frac{9}{2}} = \tan{Q}

    Q = \arctan\left(\sqrt[3]{\frac{9}{2}}\right)

     <br />
Q \approx 58.8^{\circ}<br />

    now go back and determine the length of the ladder.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Integrals with trigonometric functions
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 22nd 2011, 05:53 AM
  2. Replies: 6
    Last Post: August 29th 2010, 06:23 PM
  3. Replies: 3
    Last Post: February 23rd 2010, 05:54 PM
  4. Replies: 2
    Last Post: February 19th 2010, 11:37 AM
  5. Replies: 6
    Last Post: April 8th 2007, 12:26 PM

Search Tags


/mathhelpforum @mathhelpforum