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...

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

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We will first solve for 'p' for price. 

Subtract both sides by 0.85x.

`p + 0.85x - 0.85x = 8.5 - 0.85x`

`p = -0.85x + 8.5`

We can now write the revenue function. 

Revenue = price *amount of unit sold.

`r = (-0.85x + 8.5)(x)`

Use Distributive property.

`r = -0.85x^2 + 8.5x`

We know that Profit = revenue - cost.

So, we will have:

`P(x) = (-0.85x^2 +8.5x) - (1 + 4x) = -0.85x^2 + 4.5x - 1`

Take the derivative of both sides.

`P' = -1.7x + 4.5`

Equate it to zero, and solve for x.

`-1.7x +4.5 = 0`

`-1.7x = -4.5`

`x = 2.68 or 3000`

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