Homework Help

A producer finds that demand for his commodity obeys a linear demand equation...

user profile pic

barce90 | Student, Undergraduate | (Level 1) Honors

Posted May 11, 2013 at 8:26 PM via web

dislike 1 like

A producer finds that demand for his commodity obeys a linear demand equation p+0.85x=8.5 where p is in thousands of dollars and x is in thousands of units. If the producer's cost are given by C(x)=1+4x, what should his level of production be to maximize profits?

The level of production is= (   ) in thousands of units

1 Answer | Add Yours

user profile pic

violy | High School Teacher | (Level 3) Assistant Educator

Posted May 11, 2013 at 11:07 PM (Answer #1)

dislike 1 like

Solve for p first on the demand function to know the representation of the price. 

Subtract both sides by 0.85x on both sides.

`p = 8.5 - 0.85x`

We know the revenue = price * quantity. 

So, we will have:

`r = (8.5 - 0.85x)*x = 8.5x - 0.85x^2`

We can now write the profit function by taking note that:

Profit = Revenue - Cost

`P = 8.5x - 0.85x^2 - (1 + 4x) = 8.5x - 0.85x^2 - 1 - 4x` 

Combinbe like terms. 

`P = -0.85x^2 + 4x - 1`

Take the derivative an equate to zero.

`-1.7x + 4 = 0`

Subtract both sides by 4.

`-1.7x = -4`

Divide both sides by -1.7.

`x = 2.35 or 3` (round up).

Hence, the level of production that will maximize the profit is 3000.

 

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes