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Math Help - Find the area of a triangle.

  1. #1
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    Unhappy Find the area of a triangle.

    Find the area of a triangle enclosed by axes and tangent to y = 1/x

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  2. #2
    Junior Member MathVideos's Avatar
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    Re: Find the area of a triangle.

    Where do you run into your problem? Finding the tangent line? Also, there's an infinite amount of triangles that could be made. Do they specify anything else?
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  3. #3
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    Re: Find the area of a triangle.

    Could this picture help you?

    Find the area of a triangle.-triangle1.gif
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  4. #4
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    Re: Find the area of a triangle.

    Quote Originally Posted by joshuaa View Post
    Could this picture help you?

    Click image for larger version. 

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    Well, let's see. If the function is \displaystyle \begin{align*} y = \frac{1}{x} \end{align*}, the derivative is \displaystyle \begin{align*} \frac{dy}{dx} = -\frac{1}{x^2} \end{align*}. This tells us the gradient of the tangent line at any point x.

    So the tangent line is of the form \displaystyle \begin{align*} y = \left(-\frac{1}{x^2}\right)x + c = -\frac{1}{x} + c \end{align*}.

    c is the y intercept. To find the x intercept, we'll let y = 0, and we find that

    \displaystyle \begin{align*} 0 &= -\frac{1}{x} + c \\ \frac{1}{x} &= c \\ x &= \frac{1}{c}  \end{align*}.

    So the height of the triangle is c, the length is \displaystyle \begin{align*} \frac{1}{c} \end{align*}, which means the area is

    \displaystyle \begin{align*} A &= \frac{1}{2}(c)\left(\frac{1}{c}\right) \\ &= \frac{1}{2} \end{align*}

    So the area is \displaystyle \begin{align*} \frac{1}{2}\,\textrm{unit}^2 \end{align*}
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  5. #5
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    Re: Find the area of a triangle.

    Don't think that is correct. Example consider tangent at (1,1) Gradient is -1 Equation of tangent is y=-x+2 Intercepts are both 2 Area of triangle is 2
    In general consider tangent at (k,m) Gradient of tangent is -1/k^2 Get the equation of the tangent and the intercepts. Knowing also that km=1 wil get area of triangle=2
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  6. #6
    MHF Contributor alexmahone's Avatar
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    Re: Find the area of a triangle.

    Quote Originally Posted by Prove It View Post
    So the tangent line is of the form \displaystyle \begin{align*} y = \left(-\frac{1}{x^2}\right)x + c = -\frac{1}{x} + c \end{align*}.
    This isn't the equation of a line. (See post #5.)
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  7. #7
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    Re: Find the area of a triangle.

    Thank you very much. That was helpful
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