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Math Help - increase in rate of area of triangle question

  1. #1
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    increase in rate of area of triangle question

    please help.


    Two sides of a triangle are 4m and 5m in length and the angle between them is increasing at a rate of 0.06 radians/sec. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is pi/3 (60 degrees)
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  2. #2
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    and

    The stiffness of a rectangular beam is jointly proportional to the breadth and the cube of the depth. Find the dimensions of the stiffest beam that can be cut from a log in the shape of a right-circular cylinder of radius a centimeters.
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  3. #3
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    Quote Originally Posted by dklee41 View Post
    please help.


    Two sides of a triangle are 4m and 5m in length and the angle between them is increasing at a rate of 0.06 radians/sec. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is pi/3 (60 degrees)
    Area of a triangle is half the sine of an angle times the two connected sides. So you have: A=\frac{1}{2}(4)(5)\sin\theta

    Change the variables to get: y=\frac{1}{2}(20)\sin x=10\sin x

    Now your job is to find the slope when x=\frac{\pi}{3} can you do that?

    The graph is shown below:
    Attached Thumbnails Attached Thumbnails increase in rate of area of triangle question-area-triangle.jpg  
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  4. #4
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    what do you do with the rate of change of the angle .06 radians/sec?
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  5. #5
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    [soze=3]Hello, dklee41![/size]

    The stiffness of a rectangular beam is jointly proportional
    to the breadth and the cube of the depth.
    Find the dimensions of the stiffest beam that can be cut from a log
    in the shape of a right-circular cylinder of radius a centimeters.
    Code:
                  * * *
              *           *
            *-------+-------*
           *|       :       |*
            |      y:       |
          * |       :       | *
          * |       *       | *
          * |       : \     | *
            |      y:   \a  |
           *|       :     \ |*
            *-------+-------*
              *  x     x  *
                  * * *

    The breadth is 2x, the depth is 2y.

    Since S \:=\:kbd^3, we have: . S \:= \:k(2x)(2y)^3\:=\:16kxy^3 [1]

    From the diagram we have: . x^2 + y^2 \:=\:a^2\quad\Rightarrow\quad y \:=\:\sqrt{a^2 - x^2} [2]


    Substitute [2] into [1]: . S \;= \;16kx\left[\left(a^2-x^2\right)^{\frac{1}{2}}\right]^3 \;= \;16kx\left(a^2-x^2\right)^{\frac{3}{2}}

    And that is the function we must maximize . . .

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  6. #6
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    what is does the variable k represent? and the maximum is found by the derivative correct? how would one go about finding the derivative of the last equation?
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