Results 1 to 4 of 4

Math Help - Optimizing Problem, max/min, rectangle inside a triangle

  1. #1
    Newbie
    Joined
    Mar 2010
    Posts
    2

    Optimizing Problem, max/min, rectangle inside a triangle

    Find the area of the largest rectangle that can be inscribed in a right triangle with legs adjacent to the right angle of lengths 5cm and 12cm. The two sides of the rectangle lie along the legs.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member apcalculus's Avatar
    Joined
    Apr 2009
    From
    Boston
    Posts
    293
    Quote Originally Posted by ellenberger View Post
    Find the area of the largest rectangle that can be inscribed in a right triangle with legs adjacent to the right angle of lengths 5cm and 12cm. The two sides of the rectangle lie along the legs.
    Ellen:

    How much progress have you made with this problem?

    Let x and y be the sides.

    x will be the side of the rectangle along the triangle side of length 5, and y will be the side along the triangle side of length 12.

    Identify the right triangle (lower right if side of length 5 is the base) with legs (5-x) and y. This triangle is similar to the big right triangle, so the following ratio is valid:

    \frac{y}{12} = \frac{5-x}{5}

    Solve for y:

    y = 12 - \frac{12x}{5} (*)

    The objective function is the area of the rectangle:

    A = x y

    Write this area as a single variable function by using substitution (*) above, then apply differential calculus techniques to optimize.

    I hope this helps.

    Good luck!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2010
    Posts
    2
    Quote Originally Posted by apcalculus View Post
    Ellen:

    How much progress have you made with this problem?

    Let x and y be the sides.

    x will be the side of the rectangle along the triangle side of length 5, and y will be the side along the triangle side of length 12.

    Identify the right triangle (lower right if side of length 5 is the base) with legs (5-x) and y. This triangle is similar to the big right triangle, so the following ratio is valid:

    \frac{y}{12} = \frac{5-x}{5}

    Solve for y:

    y = 12 - \frac{12x}{5} (*)

    The objective function is the area of the rectangle:

    A = x y

    Write this area as a single variable function by using substitution (*) above, then apply differential calculus techniques to optimize.

    I hope this helps.

    Good luck!



    very helpful, thank you
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by ellenberger View Post
    very helpful, thank you
    Asked here: http://www.mathhelpforum.com/math-he...-problems.html

    Thread closed.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: June 2nd 2011, 04:56 PM
  2. Maximise a rectangle inside a triangle
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 20th 2010, 03:16 PM
  3. Triangle inside rectangle
    Posted in the Math Challenge Problems Forum
    Replies: 2
    Last Post: November 20th 2009, 05:41 PM
  4. Optimizing a rectangle
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 18th 2009, 05:55 PM
  5. Optimizing a rectangle
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 18th 2009, 04:30 PM

Search Tags


/mathhelpforum @mathhelpforum