Chapter 4.7, Problem 3E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Numerical Graphical and Analytic Analysis Find two positive numbers whose sum is 110 and whose product is a maximum.(a) Analytically complete six rows of a table such as the one below. (The first two rows arc shown.) Use the table to guess the maximum product. First Number, x Second Number Product, P 10 110 - 10 10 ( 110 − 10 ) = 1000 20 110- 20 20 ( 110 − 20 ) = 1800 (b) Write the product P as a function of x.(c) Use calculus to find the critical number of the function in part (b). Then find the two numbers.(d) Use a graphing utility to graph the function in part (b) and verify the solution from the graph.

(a)

To determine

To calculate: The estimated maximum product for two numbers with sum 110 by completing the table given below.

First number,xSecond numberP101101010(11010)=1000201102020(11020)=1800

Explanation

Given:

The table:

FirstÂ number,xSecondÂ numberP10110âˆ’1010(110âˆ’10)=100020110âˆ’2020(110âˆ’20)=1800

Calculation:

Consider a number 30. As the sum of the two numbers needs to be 110, the second number would be,

110âˆ’30=80

Then the product would be,

30(80)=2400

Do the same for number 40, 50 and 60, 70, 80, 90

(b)

To determine

To calculate: The product as a function of x where, the sum of the two numbers is 110.

(c)

To determine

To calculate: The critical points of the function P(x)=x(110x) and the numbers such thattheir product is maximum.

(d)

To determine

To graph: The function obtained in part (b) which is P(x)=x(110x) and to verify the maxima obtained.

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