Function and Optimal Announcement Time

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Mid-term exam (UIS-2012)
1/ The output of plant is Q = 0.08x2 + 0.12xy + 0.03y2 units per day, x is the no of hours skilled labor, y is the no of hours unskilled labor. Now 80 hours skilled labor and 200 hours unskilled labor. The output at a certain plant is units per day, where xis the number of hours of skilled labor used and y the number of hours of unskilled labor used. Currently, 80 hours of skilled labor and 200 hours of unskilled labor are used each day. Use calculus to estimate the change in unskilled labor that should be made to offset a 1-hour increase in skilled labor so that output will be maintained at its current level.

2/ The organizer of a sports event estimates that if the event is announced x days in advance, the revenue obtained will be R(x) thousand $ where R(x) = 400 + 120x - x2. The cost of advertising event for x days is C(x) thousand $, where C(x) = 2x2+300. a) Find profit function P(x) =R(x) – C(x) & sketch graph. b) How many days in advance should the event be announced in order to maximize profit. What is the max profit? c) What is the ratio revenue to cost Q(x) =R(x)C(x) at the optimal announcement time found in part(b). What respond when x-> 0. Interpret the result.

3/ A manufacturer can produce digital recorders at a cost of $50 apiece. It is estimated that if the recorders are sold for p dollars apiece consumers with buy q = 120 – p recorders each month. a) Express the manufacturers profit P as a function of q. b) What is the average rate of profit obtain as the level of production increase from q = 0 to q = 20. c) At what rate is profit changing when q = 20 recorders are produced? Is the profit increasing or decreasing at this level of production?

4/ It has been observed that the no of chips made by crickets each minutes depends on the temperature Crickets wont chips if the temperature is 38F or less and observations yield the following dates: No of chips ( C ) : 0 5 10 20 60...
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