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Math Help - Rectangle Inscribed in Semicircle

  1. #1
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    Rectangle Inscribed in Semicircle

    A rectangle is Inscribed in a semicircle of radius 2.
    Let P = (x, y) be the point in quadrant 1 that is a vertex of the rectangle and is on the circle.

    (a) Express the area A of the rectangle as a function of x.

    (b) Express the perimeter p of the rectangle as a function of x.
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  2. #2
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    Quote Originally Posted by symmetry View Post
    A rectangle is Inscribed in a semicircle of radius 2.
    Let P = (x, y) be the point in quadrant 1 that is a vertex of the rectangle and is on the circle.

    (a) Express the area A of the rectangle as a function of x.

    (b) Express the perimeter p of the rectangle as a function of x.
    Hello,

    to (a)

    The area of the rectangle is calculated in general by:

    A=length \times width

    According to your problem the area is:

    A=2x \cdot y. Now use Pythagoran theorem:

    x^2+y^2=r^2 \Longleftrightarrow y=\sqrt{r^2-x^2}. Thus:

    A=2x \cdot \sqrt{r^2-x^2}. That means with r = 2:

    A=2x \cdot \sqrt{4-x^2}

    to (b):

    The perimeter of a rectangle is calculated in general:

    p=2 \cdot {length} + 2 \cdot {width}. Plug in the values you know:

    p=2 \cdot 2x + 2 \cdot \sqrt{4-x^2}= 4x+ 2 \cdot \sqrt{4-x^2}
    Attached Thumbnails Attached Thumbnails Rectangle Inscribed in Semicircle-rechteck_halbkreis.gif  
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  3. #3
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    ok

    Another great reply. Tell me, how do you make those images?
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