Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 2 - Review - Modeling with Functions - Problems - Page 243: 25

Answer

(a) $A(x)=-\frac{\pi+4}{8}x^2+15x$ (b) $x\approx8.4ft$, height $\approx4.2ft$

Work Step by Step

(a) The perimeter is the sum of the semicircle and three sides of the rectangular frame, so the length of the height of the rectangular is $(30-\frac{\pi x}{2}-x)/2$, and the area of the window is $A=x(30-\frac{\pi x}{2}-x)/2+\frac{\pi (x/2)^2}{2}=-\frac{\pi+4}{8}x^2+15x$ (b) As shown in the graph, the maximum of the window area happens when $x\approx8.4ft$ and the height would be $4.2ft$
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