   Chapter 13.6, Problem 31E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Producer’s surplus If the supply function for x units of a commodity is p =   30   +   100  ln  ( 2 x +   1 ) dollars, what is the producer's surplus at x   = 30 ?

To determine

To calculate: The producer’s surplus at x=30 if supply function for x units of a commodity is p=30+100ln(2x+1) dollars.

Explanation

Given Information:

Supply function for x units of a commodity is p=30+100ln(2x+1) dollars.

Formula used:

Producer surplus at x=t is given by,

PS=p(t)t0tp(x)dx

The formula for integration by parts is given by:

udv=uvvdu

Calculation:

As it is provided that supply function for x units of a commodity is,

p=30+100ln(2x+1) …… (1)

Now, producer surplus at x=30 is given by,

PS=p(30)30030p(x)dx …… (2)

Substitute 30 for x in equation (1) to get,

p=30+100ln(60+1)=30+100ln61=441.09

Thus, substitute 441.09 for p(30) and 30+100ln(2x+1) for p(x) in equation (2) as,

PS=30(441.09)030[30+100ln(2x+1)]dx

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