Final answer:
The maximum annual profit for the production of clocks can be calculated by finding the difference between total revenue and total cost, and then determining the quantity of clocks that maximizes this profit. The revenue and cost functions need to be combined into a profit function, whose critical point within the production constraint indicates the quantity for maximum profit.
Step-by-step explanation:
To calculate the maximum annual profit for the production of clocks, we first acknowledge that profit (P) is the difference between total revenue (R) and total cost (C), or P(x) = R(x) - C(x). The given total revenue function is R(x) = 500x - 0.01x2, and the total cost function is C(x) = 160x + 100,000. The profit function thus becomes P(x) = (500x - 0.01x2) - (160x + 100,000).
Expanding the profit function, we have P(x) = 500x - 0.01x2 - 160x - 100,000 = -0.01x2 + 340x - 100,000. To find the x that maximizes profit, we can utilize calculus to find the critical points by setting the derivative of P(x) to zero and solving for x. Applying appropriate optimization techniques, we ultimately seek to find the x that provides the maximum profit within the range of 0≤x≤25,000.
Finding the derivative of the profit function P'(x) = -0.02x + 340 and setting it to zero gives us the equation 0 = -0.02x + 340, which can be solved for x. The value of x that maximizes profit can then be substituted back into the original profit function to determine the maximum annual profit, which should correspond to one of the multiple-choice options given: A) $2,990,000 B) $3,090,000 C) $2,890,000 D) $2,790,000.