Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Section 2.2 - Quadratic Functions - Exercise Set - Page 332: 76

Answer

The number of cherry trees which should be planted per acre to maximize the annual yield for the acre is $40$ and the maximum number of pounds of cherries per acre is $1600$.

Work Step by Step

The number of pounds of cherries $N$ depends on the number of cherry trees, x. In particular the number of pounds of cherries is the original number $50$, minus the number lost to overcrowding. The number of pounds of cherries per tree is the original number of pounds of cherries, $50$, minus the decrease due to overcrowding. $N\left( x \right)=50-x$ The maximum number of cherry trees is $30+x$. The number of pounds of cherries $R$ for the agency is the number of pounds of cherries per tree times the number of cherry trees $30+x$. $\begin{align} & R\left( x \right)=\left( 50-x \right)\left( 30+x \right) \\ & =1500+20x-{{x}^{2}} \end{align}$ Thus, the number of pounds of cherry is $R\left( x \right)=-{{x}^{2}}+20x+1500$. Compare it with the standard equation of quadratic function $f\left( x \right)=a{{x}^{2}}+bx+c$ The value of a is $-1$ , b is $20$ and c is $1500$ Recall that if $a<0$ , then f has a maximum that occurs at $x=\frac{-b}{2a}$ and maximum value is $f\left( \frac{-b}{2a} \right)$ Since, $a=-1$ , $a<0$ , $R$ has a maximum. Recall that the maximum value is at $x=\frac{-b}{2a}$. Substitute $-1$ for a and $20$ for b $\begin{align} & x=\frac{-b}{2a} \\ & =\frac{-20}{-2} \\ & =10 \end{align}$ The maximum number of cherry trees is $30+x$. Substitute $10$ for x. Thus, the maximum number of cherry trees is $30+x=40$ And the maximum value is $f\left( \frac{-b}{2a} \right)$. Substitute $10$ for x in $R\left( x \right)=-{{x}^{2}}+20x+1500$ $\begin{align} & R\left( x \right)=-{{x}^{2}}+20x+1500 \\ & =-\left( 100 \right)+20\left( 10 \right)+1500 \\ & =1600 \end{align}$ Hence, the maximum number of cherry trees is $40$ and maximum number of pounds of cherry is $1600$
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