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 |
⋮ | ⋮ | ⋮ |
(a)
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
(b)
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
The product of these two numbers is computed as follows,
Thus,
Differentiate P with respect to x,
For critical points,
Differentiate
P is maximum for
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.
Chapter 4 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
Additional Math Textbook Solutions
Precalculus: Mathematics for Calculus - 6th Edition
Calculus: Single And Multivariable
University Calculus
University Calculus: Early Transcendentals (4th Edition)
Thomas' Calculus: Early Transcendentals (14th Edition)
Precalculus: Mathematics for Calculus (Standalone Book)
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