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Thread: Optimization problem - rectangle inscribed in a triangle

  1. #1
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    Question Optimization problem - rectangle inscribed in a triangle

    Hi,

    I'm trying to solve for the following problem: "determine the area of the largest triangle that can be inscribed in a right triangle with legs adjacent to the right angle that are 5 cm and 12 cm". This is how I solved the problem:

    Optimization problem - rectangle inscribed in a triangle-21216293_1514262488629840_187377329_o.jpg


    I got the correct solution. However, now I'm curious whether or not you could use implicit differentiation to solve this problem? As in, whether or not we could solve this question without plugging in (5 - (5x/12)) into the equation. I have high bets that you can't, but I would just like confirmation so then I can feel more confident in how to approach optimization problems.

    - Olivia
    Last edited by otownsend; Aug 28th 2017 at 07:56 PM.
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  2. #2
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    Re: Optimization problem - rectangle inscribed in a triangle

    we could use the method of Lagrange multipliers

    Optimize xy subject to the constraint 5 x + 12 y = 60
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  3. #3
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    Re: Optimization problem - rectangle inscribed in a triangle

    By implicit differentiation:

    $A' = x\dfrac{dy}{dx} + y = 0$

    This simplifies to $\dfrac{dy}{dx} = -\dfrac{y}{x}$

    You also have $y = 5-\dfrac{5x}{12}$ so $\dfrac{dy}{dx} = -\dfrac{5}{12}$

    Setting these two expressions for $\dfrac{dy}{dx}$ equal, we have:

    $\dfrac{dy}{dx} = -\dfrac{y}{x} = -\dfrac{5}{12}$

    So, we now have a system of linear equations:

    $y=\dfrac{5x}{12}$ and $y = 5-\dfrac{5x}{12}$

    Setting them equal gives $x=6$.
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  4. #4
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    Re: Optimization problem - rectangle inscribed in a triangle

    I understand that dy/dx = -y/x... but why does y = 5- (5x/12) imply that dy/dx = -5/12? And where would does the equation y=5x/12 come from?

    Sorry I just began learning implicit differentiation.
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  5. #5
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    Re: Optimization problem - rectangle inscribed in a triangle

    Quote Originally Posted by otownsend View Post
    I understand that dy/dx = -y/x... but why does y = 5- (5x/12) imply that dy/dx = -5/12?
    For $y = 5-\dfrac{5x}{12}$, we take the derivative of both sides:

    $\begin{align*}\dfrac{d}{dx}(y) & = \dfrac{d}{dx}\left( 5 - \dfrac{5x}{12} \right) \\ \dfrac{dy}{dx} & = \dfrac{d}{dx}(5) - \dfrac{d}{dx}\left( \dfrac{5x}{12} \right) \\ \dfrac{dy}{dx} & = 0 - \dfrac{5}{12} \\ \dfrac{dy}{dx} & = -\dfrac{5}{12}\end{align*}$

    Quote Originally Posted by otownsend View Post
    And where would does the equation y=5x/12 come from?
    We now have two formulas for $\dfrac{dy}{dx}$, so we set them equal to each other. This gives $-\dfrac{y}{x} = -\dfrac{5}{12}$. We solve for $y$ by multiplying both sides by $-x$:

    $\begin{align*}-\dfrac{y}{\cancel{x}}(\cancel{-x}) & = -\dfrac{5}{12}(-x) \\ y & = \dfrac{5x}{12}\end{align*}$
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  6. #6
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    Re: Optimization problem - rectangle inscribed in a triangle

    Alright yeah that makes sense, thanks!
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