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What's the value of
h which minimize the perimeter?
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.
I found the value of w but couldn't find the value of h.
Calculus
1 Answer
Cesareo R.
Nov 21, 2017
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T ⋅ w = h ⋅ w + ⋅ 0.9 ⋅ w2
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P = w + 2 ⋅ h + 2√T 2 + ( ) = w + 2 ⋅ h + 2w√0.92 + ( ) = 2 ⋅ h
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so calling c0 =
, c1 = 2, c2 = 2√0.92 + ( ) + 1
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we have
find
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min P = c1 h + c2 w
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A = h ⋅ w + c0 w2
A − c0 w2
substituting h =
into P we get
w
c1 (A − c0 w2 )
P = c1 w +
w
and
c2 (A − c0 w2 )
dP
= − 2c0 c1 + c2 −
=0
dw
w2
giving
w0 =
√c1 A
= 0.962445√A
√ c2 − c0 c1
and consequently
h0 =
A − c0 w20
w0
= 2√Ac1 (c2 − c0 c1 ) = 0.60592√A
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