University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 1 - Section 1.1 - Functions and Their Graphs - Exercises - Page 13: 72


$a.\:\:\:\:C(x)=180\sqrt{x^2+640000}+1056000-100x$ $b.\:\:\:\:$ The least expensive point is less than $\ 2000 \mathrm{ft}\ $ from the point $\ P.$ $\:$ $ $

Work Step by Step

$a.\:\:\:$ First of all we need to find the distance from the power plant to the point $\:Q.\:$ Distance from $\:Q\:$ to $\:\mathrm{City}\:$ is $\:2(5280)-x=10560-x\:\mathrm{ft}.$ We have converted the miles to feet. Now apply the pythagorean theorem to calculate the distance across the river, let us say represented by $\:d.$ $d^2=800^2+x^2$ $d=\sqrt{x^2+640000}$ So, the total cost for laying the cable would be: $C(x)=180\cdot (\sqrt{x^2+640000})+100\cdot (10560-x)$ $C(x)=180\sqrt{x^2+640000}+1056000-100x$ $b.\:\:\:$ Pick a few numbers for $\:x\:$ and calculate the cost $\:C(x).$ $\:\:\:\:\:\:\:\mathrm{See\:the\:table\:above.}$ Based on the table, it seems that the cost is a minimum for some value of $\:x\:$ that is below $\:2000\:\mathrm{ft}.$
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