Final answer:
The correct cost functions for the cookies are $1.25 per cookie for less than or equal to 12 cookies, and $12.00 plus $0.75 for each cookie beyond the first dozen. Options A and C are the correct constraints for the cookie pricing structure provided in the question.
Step-by-step explanation:
The student is working on a problem that involves creating a cost structure based on the quantity of cookies sold. Let's identify the correct cost functions based on the given rates.
- For less than 12 cookies, the cost is $1.25 per cookie. So, the cost function is 1.25x when x ≤ 12.
- For exactly 12 cookies, the cost is a flat rate of $1.00 per cookie. Therefore, the cost is $12.00 total for 12 cookies.
- For more than 12 cookies, the first 12 cookies cost $12.00, and each additional cookie costs $0.75. So, for x cookies, where x > 12, the cost function is $12.00 plus $0.75 for each cookie beyond the first dozen. The correct function here is 12.00 + 0.75(y), where y is the number of cookies beyond the first dozen, which means y = x - 12.
Given these prices, the correct constraints for this situation out of the provided options would be:
- 1.25x when x ≤ 12 (Option A)
- $12.00 + 0.75(y) when x > 12, noting that this translates to 12.00 + 0.75(x - 12) to account for the flat rate for the first dozen (Option C)
- None of the other provided options (B, D, E) correctly describe the pricing structure
A conceptual link can be made with the law of diminishing marginal utility and cost/benefit analysis as customers buy more cookies, they may value each additional cookie less, and the cost reflects this by reducing after a certain quantity is purchased.
This also involves calculating the opportunity cost, which in this case can be understood as the additional cookies that could be purchased instead of the first dozen due to the lower rate.