Chapter 7.7, Problem 1P

To determine

Tofind: theanother number if one number is $x$ and the sum of the numbers is $20$ .

Answer to Problem 1P

The another number is $20-x$ and the product is $p\left(x\right)=20x-x^2$ and the maximum value of product is $100$ .

Explanation of Solution

Given information:

The sum of the numbers is $20$ and one number is $x$ .

Calculation: to find the another number, let $y$ is the another number,

Since, the sum of the numbers is $20$ ,

$\begin{array}{l}x+y=20\\ y=20-x\text{ [subtract]}\end{array}$

To find the product of the numbers, multiply both,

$\begin{array}{l}p\left(x\right)=x\left(20-x\right)\\ p\left(x\right)=20x-x^2\text{ [multiply]}\end{array}$

To find the maximum value of product, first differentiate and then equal to zero,

$\begin{array}{l}p\left(x\right)=20x-x^2\\ p\text{'}\left(x\right)=20-2x\text{ [differentiate]}\\ 20-2x=0\text{ [equal to zero]}\\ x=10\text{ [simplify]}\end{array}$

Now, substitute in the product function,

$\begin{array}{l}p\left(10\right)=20\left(10\right)-{\left(10\right)}^2\text{ [substitute]}\\ p\left(10\right)=200-100\text{ [multiply]}\\ p\left(10\right)=100\text{ [subtract]}\end{array}$

Thus, the maximum value of product is $100$ .