Calculus about Norman Window?

S.S.

2016-10-02 2:01 am

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?

回答 (1)

Lopez

2016-10-03 2:21 pm

✔ 最佳答案

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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