   Chapter 7.5, Problem 33E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Finding Positive Numbers In Exercises 33-36, find three positive numbers x, y, and z that satisfy the given conditions.The sum is 45 and the product is a maximum.

To determine

To calculate: The three positive numbers x, y and z if their sum is 45 and the product is a maximum.

Explanation

Given Information:

The sum of three numbers x, y and z is 45 and the product is a maximum.

Formula used:

If the function y=f(x) has a maximum at a point then their first partial derivatives fx(x) and fy(x) at that point are zero.

Calculation:

Consider the three numbers be x, y and z.

The sum of three numbers x, y and z is 45.

x+y+z=45z=45xy

The product P of the three numbers is,

P=xyz

Substitute z=45xy in the equation P=xyz,

P=xy(45xy)=45xyx2yxy2

Now, the first partial derivatives are,

Px=45y2xyy2Py=45xx22xy

Put Px=0,

45y2xyy2=0y(452xy)=0

This gives y=0, 452x=y

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