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BAX-400 – Foundations of Analytics

Calculus Homework

The shelf life (in months) of a certain drug is a continuous random variable with probability density function shown in the figure below.

Find the probability that the drug has a shelf life of

a. (5 points) Between 10 and 20 months

b. (5 points) At most 30 months

c. (15 points) Suppose the pharmacist wants to be 95% certain that the drug is still good when it is sold. How long is it safe to leave the drug on the shelf?

A manufacturer is planning to sell a new product at the price of $150 per unit and estimates that x thousand dollars is spent on development and y thousand dollars is spent on promotion, approximately units of the product will be sold. The cost of manufacturing the product is $50 per unit. If the manufacturer has a total of $8,000 to spend on development and promotion, how should this money be allocated to generate the largest possible profit?

The total cost (in dollars) of producing x food processors is

a. (10 points) Find the exact cost of producing the 21^st food processor

b. (10 points) Using marginal cost, find the approximate cost of producing the 21^st food processor.

The total profit (in dollars) from the sale of x gas grills is

a. (5 points) Find the average profit per grill if 40 grills are produced.

b. (5 points) Find the marginal average profit at a production level of 40 grills and interpret the results.

c. (5 points) Use the results from parts a and b above to estimate the average profit per grill if 41 grills are produced.

Evaluate the limit

An efficiency study of the morning shift at a certain factory indicates that an average worker who is on the job at 8:00 A.M. will have assembled f(x) = -x3 + 6x2 + 15x units x hours later. The study indicates further that after a 15-minute coffee break the worker can assemble g(x) = -3/1x3 + x2 + 23x units in x hours. Determine the time between 8:00 A.M. and noon at which a 15-minute coffee break should be scheduled so that the worker will assemble the maximum number of units by lunchtime at 12:15 P.M.

Maggie, the manager of Comfortfoot sandal company, determines that months after initiating an advertising campaign, S(t) hundred pairs of sandals will be sold, where

a. (10 points) Find S'(t) and S''(t)

b. (10 points) At what time will sales be maximized? What is the maximum level of sales?

c. (10 points) Maggie plans to end the advertising campaign when the sales rate is minimized. When does this occur? What are the sales level at this time?

A BART train arrives every 12 minutes at a particular station. If you arrive at the station at a random time (that is, with no knowledge of the train schedule), then the random variable X, representing the time you must spend waiting for the next train, is said to be uniformly distributed on the interval [0, 12]. Using numerical integration, determine the probability that you wait between 5 and 10 minutes?

A manufacturing process produces lightbulbs with life expectancies that are normally distributed with a mean of 500 hours and a standard deviation of 100 hours. Using numerical integration, determine the probability that a randomly selected light bulb is expected to last between 500 and 670 hours. Use numerical integration and not charts in the books. Show the formula used and your work.

An amusement company maintains records for each video game installed in an arcade. Suppose that C(t) and R(t) represent the total accumulated costs and revenues (in thousands of dollars), respectively, years after a particular game has been installed. Suppose also that

The value of t for which C'(t) = R'(t) is called the useful life of the game.

a. (10 points) Find the useful life of the game, to the nearest year.

b. (10 points) Find the total profit accumulated during the useful life of the game.

Public awareness of a congressional candidate before and after a successful campaign was approximated by

where t is the time in months after the campaign started and P(t) is the fraction of the number of people in the congressional district who could recall the candidate’s name.

a. (10 points) What is the average fraction of the number of people who could recall the candidate’s name during the first 7 months of the campaign?

b. (10 points) What is the average fraction of the number of people who could recall the candidate’s name during the first 2 years of the campaign?