determine the dimensions of the window which minimize the perimeter.?

👨🏻‍🏫 學術台數學

[匿名]

2017-11-14 3:01 pm

Consider a window the shape of which is a rectangle of height h surmounted by a triangle having a height T that is 0.9 times the width w of the rectangle (as shown in the figure below).

If the cross-sectional area is A, determine the dimensions of the window which minimize the perimeter.

Height:???
Width:???

更新1:

Determine the dimensions of the window which minimize the perimeter?

回答 (1)

Paul Liao

2017-11-17 11:17 pm

area, A = h*w + wT/2 = hw + 0.45 w^2
=>
h=(A - 0.45*w^2)/w

perimeter=(w+2h)+2*w*(0.5^2+0.9^2)^(1/2)
=2w+ (A-0.45*w^2)/w + 2*w*(0.5^2+0.9^2)^(1/2)
=[2-0.45+ 2*(0.5^2+0.9^2)^(1/2)] w + A/w >= 2√{A[2-0.45+ 2*(0.5^2+0.9^2)^(1/2)]}
when [1.55+ 2*(0.5^2+0.9^2)^(1/2)] w = A/w

w =√{A/ [1.55+ 2*(0.5^2+0.9^2)^(1/2)]} ~0.526 √A

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