1. The demand for a product in dollars is given by p(x) = 53/(x)^1/2 Fixed cost are $608 and the cost to produce each item is $0.53. Find the production level of x that maximizes profit within the range of 0<(or equal to)x< (or equal to)7530.
2. An efficiency study of the afternoon shift (12:00-4p.m.) at a factory shows that the average worker who starts at noon will have produced Q(t) units in t hours given by the formula: Q(t) = -t^3 + 9/2t^2 + 15t
At what time during the four-hour shift will the worker's production level reach the point of diminishing returns? At this time, how many units has the workers produced? What is the average production level for a worker over the four-hour shift?
3. An open box is made from cardboard by cutting out squares of equal size from the corners and then folding up the sides. Determine the volume of the largest box that can be constructed if the cardboard is originally 5 inches by 10 inches.
Functions are used to maximize profit, analyze diminishing returns and maximize volume. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.