# Calculus

A gutter with trapezoidal cross-section is to be made from a long sheet of stainless steel that is 30 cm wide by turning up one-fourth of the width on each side. What width across the top will give the maximum cross sectional area?

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1. So, the sloping sides are 7.5 each, and the bottom is 15cm wide. Thus, if the top has width 2x+15, we have a cross-section area

a = (2x+15 +15)/2 √((15/2)^2-x^2)
= (x+15)/2 √(225-x^2)

da/dx = -(8x^2+60x-225) / (2√225-x^2))

da/dx=0 when x=2.7452

So, the width for maxim area is 20.49cm

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