To solve the problem, we need to find the marginal profit function [tex]\(P'(x)\)[/tex] and compute [tex]\(P'(2000)\)[/tex]. We'll also interpret the results of [tex]\(P'(2000)\)[/tex].
### Step-by-Step Solution:
1. Define the revenue function [tex]\(R(x)\)[/tex] and profit function [tex]\(P(x)\)[/tex]:
The revenue function [tex]\(R(x)\)[/tex] is found by multiplying the price [tex]\(p\)[/tex] by the quantity [tex]\(x\)[/tex]:
[tex]\[ R(x) = p \times x \][/tex]
Given the demand function:
[tex]\[ p = 600 - 0.05x \][/tex]
The revenue function becomes:
[tex]\[ R(x) = x(600 - 0.05x) \][/tex]
[tex]\[ R(x) = 600x - 0.05x^2 \][/tex]
The profit function [tex]\(P(x)\)[/tex] is the difference between the revenue [tex]\(R(x)\)[/tex] and the cost [tex]\(C(x)\)[/tex]:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Given the cost function:
[tex]\[ C(x) = 0.000002x^3 - 0.03x^2 + 400x + 80,000 \][/tex]
Substitute [tex]\(R(x)\)[/tex] and [tex]\(C(x)\)[/tex] into the profit function:
[tex]\[ P(x) = (600x - 0.05x^2) - (0.000002x^3 - 0.03x^2 + 400x + 80,000) \][/tex]
Simplify the expression:
[tex]\[ P(x) = 600x - 0.05x^2 - 0.000002x^3 + 0.03x^2 - 400x - 80,000 \][/tex]
[tex]\[ P(x) = -0.000002x^3 - 0.02x^2 + 200x - 80,000 \][/tex]
2. Find the marginal profit function [tex]\(P'(x)\)[/tex]:
The marginal profit is the derivative of the profit function [tex]\(P(x)\)[/tex] with respect to [tex]\(x\)[/tex]:
[tex]\[ P'(x) = \frac{d}{dx} (-0.000002x^3 - 0.02x^2 + 200x - 80,000) \][/tex]
Differentiate each term:
[tex]\[ P'(x) = -0.000006x^2 - 0.04x + 200 \][/tex]
3. Compute [tex]\(P'(2000)\)[/tex]:
To find [tex]\(P'(2000)\)[/tex], substitute [tex]\(x = 2000\)[/tex] into the marginal profit function:
[tex]\[ P'(2000) = -0.000006(2000)^2 - 0.04(2000) + 200 \][/tex]
[tex]\[ P'(2000) = -0.000006(4000000) - 80 + 200 \][/tex]
[tex]\[ P'(2000) = -24 - 80 + 200 \][/tex]
[tex]\[ P'(2000) = 96 \][/tex]
### Interpretation of the result:
The marginal profit [tex]\(P'(2000) = 96\)[/tex] indicates that at a production level of 2000 units per week, the profit is increasing at a rate of [tex]$96 per additional unit produced. This means if the company increases production from 2000 units to 2001 units, the additional profit gained for that next unit is approximately $[/tex]96.
