Explanation It is given that the height of the metal bracket is 10 cm and the width is 6 cm. Let the height be p and the slant height be q . Hence, the total length becomes L = p + q + q = p + 2 q . Consider the right triangle with base 3 cm , the height 10 − p cm and the slant height is q. Use Pythagoras theorem and obtain the value of q as follows: q 2 = 3 2 + ( 10 − p ) 2 q = 109 − 2 p + p 2 Thus, the function of total length L can be obtained as L = p + 2 109 − 2 p + p 2 . Differentiate the function L = p + 2 109 − 2 p + p 2 with respect to p and set it to zero. d L d p = 0 d d h [ p + 2 109 − 20 p + p 2 ] = 0 1 + 2 ( − 20 + 2 p 2 109 − 2 p + p 2 ) = 0 − 20 + 2 p 109 − 2 p + p 2 = − 1 Square both sides Then obtain the quadratic equation and solve it. 4 p 2 − 80 p + 400 = 109 − 2 p + p 2 3 p 2 − 78 p + 291 = 0 p = ( − 78 ) ± ( − 78 ) 2 − 4 ( 3 ) ( 291 ) 2 ( 3 ) p ≈ 8.27 Obtain the second derivative of L as follows. d d p [ 1 + ( − 20 + 2 p 109 − 2 p + p 2 ) ] | p = 8

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

11th Edition

ISBN: 9780137554843

Author: Allyn Washington, Richard Evans

Publisher: PEARSON+

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Chapter 27, Problem 108RE

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Chapter 27, Problem 107RE

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Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

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