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 331: 62

Answer

The required pair of numbers is $\left( 10,10 \right)$ and their product is $\underline{100}$.

Work Step by Step

Let us assume the two numbers be x and y. Then, $x+y=20$ ….. (I) Now, $\begin{align} & x+y=20 \\ & y=20-x \\ \end{align}$ And calculate xy, $\begin{align} & xy=x\left( 20-x \right) \\ & xy=20x-{{x}^{2}} \\ \end{align}$ Which is a downwards opening parabola, which attains its maximum value at $\frac{-b}{2a}$ , where $b=20$ and $a=-1$. $\begin{align} & x=\frac{-20}{-2} \\ & x=10 \\ \end{align}$ Putting the value of $x$ in the equation (I) we get, $\begin{align} & 10+y=20 \\ & y=20-10 \\ & y=10 \end{align}$ The maximum product is $10\times 10=100$. Hence, the product is maximum when both numbers are 10 and the maximum product is 100.
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