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Sagot :
To find the profit function [tex]\( P(x) \)[/tex], we need to subtract the cost function [tex]\( C(x) \)[/tex] from the revenue function [tex]\( R(x) \)[/tex].
Given the functions:
[tex]\[ C(x) = 500x^2 + 100x \][/tex]
[tex]\[ R(x) = -0.6x^3 + 700x^2 - 400x + 300 \][/tex]
The profit function [tex]\( P(x) \)[/tex] is determined as follows:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex] into the equation:
[tex]\[ P(x) = (-0.6x^3 + 700x^2 - 400x + 300) - (500x^2 + 100x) \][/tex]
Now, distribute the negative sign across the terms in [tex]\( C(x) \)[/tex]:
[tex]\[ P(x) = -0.6x^3 + 700x^2 - 400x + 300 - 500x^2 - 100x \][/tex]
Combine like terms:
[tex]\[ P(x) = -0.6x^3 + (700x^2 - 500x^2) + (-400x - 100x) + 300 \][/tex]
[tex]\[ P(x) = -0.6x^3 + 200x^2 - 500x + 300 \][/tex]
Thus, the profit function is:
[tex]\[ P(x) = -0.6x^3 + 200x^2 - 500x + 300 \][/tex]
The correct choice is:
[tex]\[ \boxed{B. \, P(x) = -0.6x^3 + 200x^2 - 500x + 300} \][/tex]
Given the functions:
[tex]\[ C(x) = 500x^2 + 100x \][/tex]
[tex]\[ R(x) = -0.6x^3 + 700x^2 - 400x + 300 \][/tex]
The profit function [tex]\( P(x) \)[/tex] is determined as follows:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex] into the equation:
[tex]\[ P(x) = (-0.6x^3 + 700x^2 - 400x + 300) - (500x^2 + 100x) \][/tex]
Now, distribute the negative sign across the terms in [tex]\( C(x) \)[/tex]:
[tex]\[ P(x) = -0.6x^3 + 700x^2 - 400x + 300 - 500x^2 - 100x \][/tex]
Combine like terms:
[tex]\[ P(x) = -0.6x^3 + (700x^2 - 500x^2) + (-400x - 100x) + 300 \][/tex]
[tex]\[ P(x) = -0.6x^3 + 200x^2 - 500x + 300 \][/tex]
Thus, the profit function is:
[tex]\[ P(x) = -0.6x^3 + 200x^2 - 500x + 300 \][/tex]
The correct choice is:
[tex]\[ \boxed{B. \, P(x) = -0.6x^3 + 200x^2 - 500x + 300} \][/tex]
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