Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 1 - Functions and Limits - 1.2 Mathematical Models: A Catalog of Essential Functions - 1.2 Exercises - Page 34: 18


(a) $$C=13N+900$$ The image of the graph has been added. (b) The slope of the graph is $13$. It represents the rate of change of the cost with respect to the number; for example, if the number changes by unit, then the cost changes by $13$. (c) The $C$-intercept of the graph is $900$. It represents the value of $C$ for $N=0$.

Work Step by Step

To find the equation of this linear function, $C=mN+d$, we must compute the slope, $m$, and the $C$-intercept, $d$. For the line passing through the points, $(x_1,y_1)$ and $(x_2, y_2)$, $m=\frac{y_1-y_2}{x_1-x_2}$. So we have $$m= \frac{4800-2200}{300-100}=13.$$To compute the $C$-intercept, one can put one of the given points, for example $(100,2200)$ into the line equation. So we have $$C= 13N+d \quad \Rightarrow \quad 2200=13 \cdot 100 +d \quad \Rightarrow \quad d=900.$$ So the the function is of the form $$C=13N+900.$$
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