Let's analyze the given cost model [tex]\(C(x)\)[/tex], sales price model [tex]\(S(x)\)[/tex], revenue model [tex]\(R(x)\)[/tex], and profit, then derive the final expression:
1. Given Cost Model: [tex]\[ C(x) = 0.18 x^3 + 0.02 x^2 + 4 x + 180 \][/tex]
2. Given Sale Price Model: [tex]\[ S(x) = 95.4 - 6x \][/tex]
3. Given Revenue Model: [tex]\[ R(x) = x \cdot S(x) = x \cdot (95.4 - 6x) = 95.4x - 6x^2 \][/tex]
4. Profit Model: Profit is given by the difference between revenue and cost:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substituting [tex]\(R(x)\)[/tex] and [tex]\(C(x)\)[/tex] into the profit formula:
[tex]\[ P(x) = (95.4x - 6x^2) - (0.18x^3 + 0.02x^2 + 4x + 180) \][/tex]
Now let's distribute and combine like terms:
[tex]\[ P(x) = 95.4x - 6x^2 - 0.18x^3 - 0.02x^2 - 4x - 180 \][/tex]
Combine the terms:
[tex]\[ P(x) = -0.18x^3 - 6x^2 - 0.02x^2 + 95.4x - 4x - 180 \][/tex]
[tex]\[ P(x) = -0.18x^3 - (6x^2 + 0.02x^2) + (95.4x - 4x) - 180 \][/tex]
[tex]\[ P(x) = -0.18x^3 - 6.02x^2 + 91.4x - 180 \][/tex]
Therefore, the profit function is:
[tex]\[ P(x) = -0.18x^3 - 6.02x^2 + 91.4x - 180 \][/tex]
Reviewing the options given in the question, we see that this matches option (3):
(1) [tex]\( 0.18 x^3 + 6.02 x^2 + 91.4 x + 180 \)[/tex]
(2) [tex]\( 0.18 x^3 - 5.98 x^2 - 91.4 x + 180 \)[/tex]
(3) [tex]\( -0.18 x^3 - 6.02 x^2 + 91.4 x - 180 \)[/tex] [tex]\(\rightarrow\)[/tex] This matches our result
(4) [tex]\( 0.18 x^3 + 5.98 x^2 + 99.4 x + 180 \)[/tex]
Thus, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
