2. Find the half-life of a radioactive material if after 1 year 99.57% of the initial amount remains. 3. 14C dating assumes that the carbon dioxide on the Earth today has the same radioactive content as it did centuries ago. If this is true, then the amount of 14C absorbed by a tree that grew several centuries ago should be the same as the amount of 14C absorbed by a similar tree today. A piece of ancient charcoal contains only 15% as much of the radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal? (The half-life of 14C is 5715 years.) 4. The number of squirrels on the UCD campus increases continuously at a rate proportional to the number present. There are 150 present at a given time and 450 present 5 hours later. (a) How many will there be 10 hours after the initial time? (b) How long will it take for the population to double. (c) Does the answer to part (b) depend on the starting time? Explain your reasoning. 5. The cumulative sales S (in thousands of units) of a new product after it has been on the market for t years are modeled by S = Cek/t . During the first year, 5000 units were sold. The saturation point for the market is 30,000 units. That is, the limit of S as t is 30,000. (a) Solve for C and k in the model. (b) How many units will be sold after 5 years? 6. A small business assumes that the demand function for one of its new products can be modeled by p = Cekx. When p = $45, x = 1000 units, and when p = $40, x = 1200 units. Solve for C and k. 7. The management of a factory finds that the maximum number of units a worker can produce in a day is 30. The learning curve for the number of units N produced per day after a new employee has worked t days is modeled by N = 30(1 e kt). After 20 days on the job, a worker is producing 19 units in a day. How many days should pass before this worker is producing 25 units per day?