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
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`