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posted by  Novaworks on 11/2/2009 8:46:49 PM  |  status: Closed  |  Earned Karma: 1145

Find two numbers

Course Textbook Chapter Problem Needs by
Calculus N/A N/A N/A 11/2/2009 at 1:00:00 PM
Question Details:
Find two numbers whose difference is 132 and whose product is a minimum.
1? (smaller number)
2? (larger number)
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Oracle
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posted by Altair on 11/2/2009 9:05:52 PM  |  status: Live
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Response Details:
let the two number is x and y
product D = x-y = 132, =>x = 132+y
P = xy = (132+y)y = y2 + 132y
P'  = 2y+132
let P' = 0 =>2y+132 = 0
=>y = -132/2 = -66
x = 132+y = 132-66 = 66
the number is:
1: -66
2: 66

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Oracle
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posted by BEL (MNV) on 11/2/2009 9:06:36 PM  |  status: Live
Asker's Rating: Helpful   
Response Details:
let the numbewrs be X & Y
Giventhat X-Y=132 ==>Y=X-132
Produt P =XY = X (X-132)=X2 -132X
==>P'=2X-132   also  P'=0 ==>2X-132=0 ==>X=66
==>P" =2>0 ==>at X=66 product is minimum
X=66==> Y=66-132=-66
so the numbers are X=66 ,Y = - 66
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posted by hado on 11/2/2009 9:11:18 PM  |  status: Live
Asker's Rating: Helpful   
Response Details:
Let x and y be the unknown numbers

Then

y-x=132

==>

y=x+132

The product of the numbers

P=xy=x(x+132)=x2 +132x

We take the first derivative with respect to x

P'(x)=2x+132

Set P'(x)=0 ==> 2x+132=0 ==> x=-132/2=-66

and y=132+x=132-66=66

At x=-66, P(x) has local minimum by the First Derivative Test

Therefore

Smaller number: -66

Larger number: 66
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