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Math Help - Optimization Problem

  1. #1
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    Optimization Problem

    I am really stuck with this question:

    A rectangle is drawn inside a semicircle of base radius 10cm.

    a) Show that the area of the rectangle is 200 sin x cos x cm^2

    b) Calculate the maximum value of this area

    There is no diagram but there is actually a triangle inside of the rectangle with an angle of x. The base of 10cm is on the adjacent side of the triangle (so at the bottom)

    If someone can help that would be very grateful

    Thanks
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  2. #2
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    Re: Optimization Problem

    it's not hard to answer first question, using trig. ratio

    But I can answer your second question.

    You need to find the derivative of 200 sin x cos x first.
    200 sin x cos x = 100sin 2x (apply formula sin 2x = 2sin x cos x)
    dy/dx 100 sin 2x = 200cos 2x (the derivative of sin x is cos x)
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  3. #3
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    Re: Optimization Problem

    Quote Originally Posted by tq1088 View Post
    I am really stuck with this question:

    A rectangle is drawn inside a semicircle of base radius 10cm.

    a) Show that the area of the rectangle is 200 sin x cos x cm^2

    b) Calculate the maximum value of this area

    There is no diagram but there is actually a triangle inside of the rectangle with an angle of x. The base of 10cm is on the adjacent side of the triangle (so at the bottom)

    If someone can help that would be very grateful

    Thanks
    Suppose that the rectangle was drawn inside a semicircle of \displaystyle 1\,\textrm{cm} radius. Then the distance from the midpoint of the base of the rectangle (the centre of the circle) to the edge of the rectangle is \displaystyle \cos{x}\,\textrm{cm}, which means that the length of the rectangle is double this length, so \displaystyle 2\cos{x}\,\textrm{cm}. The width of the rectangle is \displaystyle \sin{x}\,\textrm{cm}.

    But this is a semicircle of radius \displaystyle 10\,\textrm{cm}, which means that each length has been magnified by a factor of \displaystyle 10. So the length of the rectangle is \displaystyle 20\cos{x}\,\textrm{cm} and the width is \displaystyle 10\sin{x}\,\textrm{cm}.

    Therefore the area of the rectangle is \displaystyle 20\cos{x}\,\textrm{cm} \times 10\sin{x}\,\textrm{cm} = 200\cos{x}\sin{x}\,\textrm{cm}^2.
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  4. #4
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    Re: Optimization Problem

    Quote Originally Posted by tq1088 View Post
    I am really stuck with this question:

    A rectangle is drawn inside a semicircle of base radius 10cm.

    a) Show that the area of the rectangle is 200 sin x cos x cm^2

    b) Calculate the maximum value of this area

    There is no diagram but there is actually a triangle inside of the rectangle with an angle of x. The base of 10cm is on the adjacent side of the triangle (so at the bottom)

    If someone can help that would be very grateful

    Thanks
    Questions like part (b) typically require calculus, as has already been suggested.

    However, it can be done without calculus by using the double angle formula.

    Unfortunately, you give no clue as to what your background is and what technique you're probably expected to use. For all I know, you don't know calculus, or you don't know the double angle formula ....

    For what it's worth, using the double angle formula the area can be written as A = 100 sin(2x) and it should be obvious what the maximum of this is. Furthermore, it should be obvious that the value of x that gives this maximum value is x = pi/4 ....
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