# To construct : A table of values, so that the sum of the numbers in the first two columns is always 23 and to find two numbers whose product is maximum. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 4.6, Problem 1E

(a)

To determine

## To construct: A table of values, so that the sum of the numbers in the first two columns is always 23 and to find two numbers whose product is maximum.

Expert Solution

The numbers are 11 and 12 that yields the maximum product that is 132.

### Explanation of Solution

Construction:

Obtain a table of values, so that the sum of the numbers in the first two columns is always 23 and their product is in third column.

 First Number Second Number Product 1 22 22 2 21 42 3 20 60 4 19 76 5 18 90 6 17 102 7 16 112 8 15 120 9 14 126 10 13 130 11 12 132

The product of the numbers of the first column and second column is written in the third column for all possible pair of numbers whose sum is 23.

From the table, by comparing the product of all pair of numbers, it can be concluded that the pair (11,12), whose sum is 23, yields the maximum product that is 132.

(b)

To determine

### To Compare: The table values in part (a) using Calculus

Expert Solution

The numbers are 11 and 12.

### Explanation of Solution

Calculation:

Let the number be x,

Given that, the sum of the two numbers is 23. So take the another number is 23x .

The product of these two numbers is computed as follows,

P=x(23x)=23xx2

Thus, P=23xx2 .

Differentiate P with respect to x,

dPdx=232x

For critical points, dPdx=0 .

dPdx=0232x=023=2xx=11.5

Differentiate dPdx with respect to x,

d2Pdx2=2<0

P is maximum for x=11.5 .

Since the value of x is 11.5, the value of x can be both 11 and 12

The product P is maximum for both 11 and 12.

Thus, the numbers are 11 and 12 with product 132.

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