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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Answer to a math 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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