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S.S.
Lv 7
S.S. 發問於 科學及數學數學 · 4 年前

Calculus about Norman Window?

A Norman window has the shape of a rectangle surmounted by a semicircle. If a Norman window is to have a perimeter of 28 meters, what should its dimensions be in order to allow the maximum amount of light through the window?

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  • Lopez
    Lv 5
    4 年前
    最愛解答

    Let the radius of semicircle be R , and the dimension of rectangle be 2R*y

    28 = πR + 2R + 2y

    2y = 28 - πR - 2R

    A

    = (1/2)πR² + 2R*y

    = (1/2)πR² + R*2y

    = (1/2)πR² + R*( 28 - πR - 2R )

    = ( π/2 - 2 - π )R² + 28R

    = ( -π/2 - 2 )R² + 28R , a parabola concave down

    dA/dR

    = ( -π/2 - 2 )*2R + 28

    = - (π+4)R + 28

    = 0

    R = 28/(π+4)

    y

    = (1/2)( 28 - πR - 2R )

    = 14 - (1/2)(π+2)R

    = 14 - (1/2)(π+2)28/(π+4)

    = 14 - (14π+28)/(π+4)

    = ( 14π + 56 - 14π - 28 )/(π+4)

    = 28/(π+4)

    Ans:

    The radius of semicircle is R , and the dimension of rectangle is 2R*y ,

    where R = y = 28/(π+4) meters

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