Page 1 of 2 12 LastLast
Results 1 to 15 of 18
Like Tree10Thanks

Thread: Calculus

  1. #1
    Senior Member Vinod's Avatar
    Joined
    Sep 2011
    From
    I live here
    Posts
    299
    Thanks
    5

    Calculus

    Hello Forumites,
    A farmer wishes to fence a field bordering a straight stream with 1000 yd. of fencing material. It is not necessary to fence the side bordering the stream. Now, i want to find what is the maximum area of a rectangular field that can be fenced in this way?

    Solution. I don't have any idea to solve this problem. I know the area of rectangle=length*width. Now, for maxima, first derivative=0 and second derivative <0. Answer provided to me is 125000 sq.yd. If any forumite know the answer, he may reply with it.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    6,189
    Thanks
    2638

    Re: Calculus

    Assume all units are yds or sq yds as appropriate.

    $l + 2w = 1000$

    $A = l w$

    $A = (1000-2w)w = 1000w - 2w^2$

    $\dfrac{dA}{dw} = 1000 - 4w$

    solve for $\dfrac{dA}{dw}=0$

    $4w = 1000$

    $w = 250$

    $l = 1000 - 2(250) = 500$

    $A = (250)(500) = 125000$
    Thanks from Vinod, topsquark and MarkFL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    2,096
    Thanks
    806

    Re: Calculus

    I would let $\displaystyle x$ be the length of the sides perpendicular to the stream and $\displaystyle y$ be the length of the side parallel to the stream. Let the total amount of fence be $\displaystyle F$.

    The area $\displaystyle A$ of the enclosed rectangle gives us the objective function:

    $\displaystyle A(x,y)=xy$

    Subject to the constraint:

    $\displaystyle g(x,y)=2x+y-F=0$

    Using Lagrange multipliers, there arises the system:

    $\displaystyle y=\lambda(2)$

    $\displaystyle x=\lambda(1)$

    This implies:

    $\displaystyle y=2x$

    Substituting into the constraint, we obtain:

    $\displaystyle 4x=F\implies x=\frac{F}{4},\,y=\frac{F}{2}$

    We then find:

    $\displaystyle A\left(\frac{F}{4},\frac{F}{2}\right)=\frac{F^2}{8 }$

    As $\displaystyle F(0,F)=F\left(\frac{F}{2},0\right)=0$, we may then conclude:

    $\displaystyle A_{\max}=\frac{F^2}{8}$

    Using the given value for $\displaystyle F$, we find:

    $\displaystyle A_{\max}=\frac{(1000\text{ yd})^2}{8}=125000\text{ yd}^2$

    I know you are looking for a solution using functions of one variable, but this should easily give you enough to do so. Solve the constraint for one of the variable, and then you can express the area in terms of that variable, and the parameter $\displaystyle F$.
    Last edited by MarkFL; Jul 26th 2018 at 09:22 PM.
    Thanks from topsquark and Vinod
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    1,909
    Thanks
    352

    Re: Calculus

    On a timed test, with a few seconds to go,
    easy for an unsuspecting(!) student to reason
    "since square produces maximum area,
    then 333 1/3...."
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Dec 2016
    From
    Earth
    Posts
    186
    Thanks
    82

    Re: Calculus

    Quote Originally Posted by DenisB View Post
    On a timed test, with a few seconds to go,
    easy for an unsuspecting(!) student to reason
    "since square produces maximum area,
    then 333 1/3...."
    How would an unsuspecting student come up with that!?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Nov 2010
    Posts
    3,576
    Thanks
    1438

    Re: Calculus

    Quote Originally Posted by greg1313 View Post
    How would an unsuspecting student come up with that!?
    That's a good question. Perhaps we should ask my students who came up with that back when I taught Calc I.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    11,213
    Thanks
    755
    Awards
    1

    Re: Calculus

    Quote Originally Posted by DenisB View Post
    ...333 1/3...."
    Isn't that an old record player setting?

    -Dan
    Thanks from MarkFL
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    2,096
    Thanks
    806

    Re: Calculus

    Quote Originally Posted by topsquark View Post
    Isn't that an old record player setting?

    -Dan
    It was 33 1/3. I recall 16, 33 1/3, 45 and 78. My dad had some 78's, but all I ever had were 33 1/3 and 45.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    11,213
    Thanks
    755
    Awards
    1

    Re: Calculus

    Quote Originally Posted by MarkFL View Post
    It was 33 1/3. I recall 16, 33 1/3, 45 and 78. My dad had some 78's, but all I ever had were 33 1/3 and 45.
    I know. I was just messing around. I even remember 8 tracks. And my ex-father-in-law had a reel to reel.

    -Dan
    Thanks from romsek and MarkFL
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    1,909
    Thanks
    352

    Re: Calculus

    I remember one of these:
    https://www.ebay.com/bhp/victrola-crank

    My dad used to listen to this on it:

    Jambalaya (On the Bayou)

    Jambalaya, crawfish pie and fillet gumbo
    For tonight, I'm-a gonna see my ma cher a mi-o
    Pick guitar, fill fruit jar and be gay-o
    Son of a gun, we'll have big fun on the bayou…
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    2,096
    Thanks
    806

    Re: Calculus

    Quote Originally Posted by topsquark View Post
    I know. I was just messing around. I even remember 8 tracks. And my ex-father-in-law had a reel to reel.

    -Dan
    I had friends with 8-tracks in their cars...I was a cassette lad myself during that period. One of my uncles, a real audiophile, had a reel to reel system. I still think analog music sounds more vibrant than digital.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor MarkFL's Avatar
    Joined
    Dec 2011
    From
    St. Augustine, FL.
    Posts
    2,096
    Thanks
    806

    Re: Calculus

    Quote Originally Posted by DenisB View Post
    I remember one of these:
    https://www.ebay.com/bhp/victrola-crank

    My dad used to listen to this on it:

    Jambalaya (On the Bayou)

    Jambalaya, crawfish pie and fillet gumbo
    For tonight, I'm-a gonna see my ma cher a mi-o
    Pick guitar, fill fruit jar and be gay-o
    Son of a gun, we'll have big fun on the bayou…
    My dad was a HUGE Hank fan...I know all his songs by heart.
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    11,213
    Thanks
    755
    Awards
    1

    Re: Calculus

    Quote Originally Posted by DenisB View Post
    I remember one of these:
    https://www.ebay.com/bhp/victrola-crank

    My dad used to listen to this on it:

    Jambalaya (On the Bayou)

    Jambalaya, crawfish pie and fillet gumbo
    For tonight, I'm-a gonna see my ma cher a mi-o
    Pick guitar, fill fruit jar and be gay-o
    Son of a gun, we'll have big fun on the bayou…
    How about
    Chicken in the bread pan peckin' out dough,
    Granny does your dog bite? No, child, no.
    (That's the Devil Went Down to Georgia version anyway.)

    -Dan
    Follow Math Help Forum on Facebook and Google+

  14. #14
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    1,909
    Thanks
    352

    Re: Calculus

    ...and why did you guys "pick on my 333 1/3" anyway ?
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    11,213
    Thanks
    755
    Awards
    1

    Re: Calculus

    Quote Originally Posted by DenisB View Post
    ...and why did you guys "pick on my 333 1/3" anyway ?
    I have to have a reason??

    -Dan
    Follow Math Help Forum on Facebook and Google+

Page 1 of 2 12 LastLast

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Dec 13th 2011, 09:11 PM
  2. Replies: 2
    Last Post: Jun 25th 2010, 10:41 PM
  3. Replies: 1
    Last Post: Feb 11th 2010, 07:09 AM
  4. Replies: 1
    Last Post: Jun 23rd 2008, 09:17 AM
  5. Replies: 1
    Last Post: Jun 7th 2008, 11:47 AM

/mathhelpforum @mathhelpforum