Answer:
Step-by-step explanation:
The cost function = C(x)
The demand function = P(x)
C(x) = 11000 + 500x - 3.6x^2 + + 0.004x^3
P(x) = 1700 - 9x
Differentiate C(x) with respect to x
C'(x) = 500 - 7.2x + 0.012x^2
C'(x) is the marginal cost
Revenue = x. P(x)
R(x) = x( 1700 -9x)
= 1700x - 9x^2
Differentiate R(x) with respect to x
R'(x) = 1700 - 18x
R'(x) is the marginal revenue
Profit is maximized when R'(x) = C'(x)
1700 - 18x = 500 - 7.2x + 0.012x^2
Collect like terms
0 = 500 - 1700 - 7.2x +18x +0.012x^2
0 = -1200 + 10.8x +0.012x^2
0 = 0.012x^2 + 10.8x - 1200
Using x=( -b +_ √b^2 - 4ac) /2a
a = 0.012 , b= 10.8 , c= -1200
x= (-10.8 +_ √(10.8^2) - 4*0.012*(-1200)) /2*0.012
= ( -10.8 +_√116.64 + 57.6) / 0.024
= (-10.8 +_ √174.24) / 0.024
= (-10.8 +_13.2) / 0.024
= (-10.8+13.2)/0.024 or (-10.8 - 13.2)/0.024
= 2.4/0.024 or -24/0.024
x= 100 or -1000
Since our profit cannot be negative, the profit = $100
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