Question

21. (Maximum profit) A company sells all the units it produces for $4 each. Company C's total cost of producing x units is given in dollars times C = 50 + 1.3x + 0.001x2 a) Write the expression for total utility P as a function of x. b) Determine the production volume x so that profit P is maximum. c) What is the value of maximum utility?

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a) Total revenue is R = 4x

Total cost isC = 50 + 1.3x + 0.001x^2

Profit is given byP = R - C = 4x - (50 + 1.3x + 0.001x^2)

P = 4x - 50 - 1.3x - 0.001x^2

P = 2.7x - 50 - 0.001x^2

b) To find the value of x that maximizes profit, take the first derivative and set it equal to zero:

\frac{dP}{dx} = 2.7 - 0.002x = 0

Solve for:

2.7 = 0.002x

x = \frac{2.7}{0.002}

x = 1350

c) Substitute back into the profit function:

P = 2.7(1350) - 50 - 0.001(1350)^2

P = 3645 - 50 - 1822.5

P = 1250

So, the maximum utility is 1250 and the production volume that maximizes it is 1350 units.

Total cost is

Profit is given by

b) To find the value of x that maximizes profit, take the first derivative and set it equal to zero:

Solve for:

c) Substitute back into the profit function:

So, the maximum utility is 1250 and the production volume that maximizes it is 1350 units.

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