Chapter 4.6, Problem 1E

Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum.

(a) Make a table of values, like the one at the right, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in your table, estimate the answer to the problem.

(b) Use calculus to solve the problem and compare with your answer to part (a).

First number	Second number	Product
1	22	22
2	21	42
3	20	60
⋮	⋮	⋮

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.

Answer to Problem 1E

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 $\left(11,12\right),$ whose sum is 23, yields the maximum product that is 132.

To determine

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

Answer to Problem 1E

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 $23-x$.

The product of these two numbers is computed as follows,

$\begin{array}{c}P=x\cdot \left(23-x\right)\\ =23x-x^2\end{array}$

Thus, $P=23x-x^2$.

Differentiate P with respect to x,

$\frac{dP}{dx}=23-2x$

For critical points, $\frac{dP}{dx}=0$.

$\begin{array}{l}\frac{dP}{dx}=0\\ 23-2x=0\\ 23=2x\\ x=11.5\end{array}$

Differentiate $\frac{dP}{dx}$ with respect to x,

$\frac{d^{2}P}{d{x}^2}=-2<0$

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.