1. A local copy center needs to buy white paper and yellow paper. They can buy from three suppliers. Supplier 1 sells a package of 20 reams of white and 10 reams of yellow for $60. Supplier 2 sells a package of 10 reams of white and 10 reams of yellow for $40. Supplier 3 sells a package of 10 reams of white and 20 reams of yellow for $50. The copy center needs 350 reams of white and 400 reams of yellow. Using Python, determine (1) how many packages they should buy from each supplier in order to minimize cost and (2) the minimum cost.
2. A new test has been developed to detect a particular type of cancer. A medical researcher selects a random sample of 1,000 adults and finds (by other means) that 4% have this type of cancer. Each of the 1,000 adults is given the new test and it is found that the test indicates cancer in 99% of those who have it and in 1% of those who do not. Based on these results, what is the probability of a randomly chosen person having cancer given that the test indicates cancer? What is the probability of a person having cancer given that the test does not indicate cancer? Round the probabilities to four decimal places.
3. If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then the volume of the water remaining in the tank after minutes is given by . Using Python, determine the rate at which water is draining from the tank. When will it be draining the fastest?
4. A rectangular container with a volume of 475 ft3 is to be constructed with a square base and top. The cost per square foot for the bottom is $0.20, for the top is $0.10, and for the sides is $0.015. Find the dimensions of the container that minimize the cost. Round to two decimal places.
5. Assume the total revenue from the sale of items is given by while the total cost to produce items is . Find the approximate number of items that should be manufactured so that profit is maximized. Justify that the number of items you found gives you the maximum and state what the maximum profit is.
6. For the following function, determine the domain, critical points, intervals where the function is increasing or decreasing, inflection points, intervals of concavity, intercepts, and asymptotes where applicable. Use this information and Python to graph the function.
7. The rate of growth of the profit (in millions) from an invention is approximated by the function where represents time measured in years. The total profit in year two that the invention is in operation is $25,000. Find the total profit function. Round to three decimals.
8. For a certain drug, the rate of reaction in appropriate units is given by where is time (in hours) after the drug is administered. Find the total reaction to the drug from 1 to 8 hours after it is administered. Round to two decimal places.
9. Determine if the following function is a probability density function on .
If it is a probability density function, then calculate and round to four decimal places. If it is not, explain why.
10. Researchers have shown that the number of successive dry days that occur after a rainstorm for a particular region is a random variable that is distributed exponentially with a mean of 9 days. Using Python, determine the (separate) probabilities that 13 or more successive dry days occur after a rainstorm, and fewer than 2 dry days occur after a rainstorm. Round the probabilities to four decimal places.