Estimating Change in Profit for a Lumber Company

Question:

What is the estimated change in profit during the third month for a lumber company that produces x units of a product with varying production quantities based on time?

Answer:

The change in profit during the third month is estimated to be 135 dollars.

Explanation:

The given data includes:

Cost function: C(x) = 0.1x^2 + 2260x + 1450

Price function: p(x) = 65(125 – 0.2x)

Quantity produced in month t: x(t) = –0.35(t – 4)^2 + 197

To calculate the change in profit during the third month:

1. Revenue in month t is given by R(t) = 65(125 – 0.2x(t)) * x(t) = 7625 – 13x(t)^2

2. Profit in month t is given by P(t) = R(t) – C(x(t)) = –13x(t)^2 + 2035x(t) – 3175

3. Marginal profit in month t is P'(t) = –26x(t) + 2035

At t = 2:

Quantity produced, x(2) = –0.35(2 – 4)^2 + 197 = 174

Marginal profit at t = 2, P'(2) = –26(174) + 2035 = 135

Therefore, the change in profit during the third month is:

P(3) – P(2) = (–13(174)^2 + 2035(174) – 3175) – (–13(174)^2 + 2035(174) – 3175) = 135 dollars

So, the estimated change in profit during the third month is $135.

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