A metal rain gutter is to have 3−inch and a 3−inch bottom the sides making an equal angle θ with the bottom. What should θ be in order to maximize carrying capasity of the gutter ?


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Question Number 130582 by bramlexs22 last updated on 27/Jan/21

$A metal rain gutter is to have 3 - inch and$ $a 3 - inch bottom the sides making an equal$ $angle θ with the bottom . What should$ $θ be in order to maximize carrying capasity$ $of the gutter ?$
	Answered by EDWIN88 last updated on 27/Jan/21

	$T h e c a r r y i n g c a p a s i t y o f t h e g u t t e r$ $i s m a x i m i z e d w h e n t h e a r e a$ $o f t h e v e r t i c a l e n d o f t h e g u t t e r$ $i s m a x i m i z e d .$ $A = 3 (3 \sin θ) + 2 (\frac{1}{2}) (3 \cos θ) (3 \sin θ)$ $A = 9 \sin θ + 9 \cos θ \sin θ$ $\frac{d A}{d θ} = 9 (2 \cos^{2} θ + \cos θ - 1) = 0$ $w h e n θ = \frac{π}{3} \Rightarrow A = \frac{27 \sqrt{3}}{4} \approx 11.7$ $T h e c a r r y i n g c a p a s i t y i s m a x$ $w h e n θ = \frac{π}{3}$
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