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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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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.
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