Oxygen purity—diminishing returns Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill is given by
(a) Find how long it will be before the oxygen level reaches its minimum.
(b) Find how long it will be before the rate of change of P is maximized. That is, find the point of diminishing returns.
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Chapter 10 Solutions
Mathematical Applications for the Management, Life, and Social Sciences
- Drug Concentration When a drug is administered orally, it takes some time before the blood concentration reaches its maximum level. After that time, concentration levels decrease. When 500 milligrams of procainamide is administered orally, one model for a particular patient gives blood concentration C, in milligrams per liter, after t hours as C=2.65(e0.2te2t) What is the maximum blood-level concentration, and when does that level occur?arrow_forwardDecay of Litter Litter such as leaves falls to the forest floor, where the action of insects and bacteria initiates the decay process. Let A be the amount of litter present, in grams per square meter, as a function of time t in years. If the litter falls at a constant rate of L grams per square meter per year, and if it decays at a constant proportional rate of k per year, then the limiting value of A is R=L/k. For this exercise and the next, we suppose that at time t=0, the forest floor is clear of litter. a. If D is the difference between the limiting value and A, so that D=RA, then D is an exponential function of time. Find the initial value of D in terms of R. b. The yearly decay factor for D is ek. Find a formula for D in term of R and k. Reminder:(ab)c=abc. c. Explain why A=RRekt.arrow_forwardPopulation Growth The game commission introduces 100 deer into newly acquired state game lands. The population N of the herd is modeled by N=20(5+3t)1+0.04t, t0 where t is the time in years. (a) Use a graphing utility to graph this model. (b) Find the populations when t=5, t=10 , and t=25. (c) What is the limiting size of the herd as time increases?arrow_forward
- Water Flea F. E Smith has reported on population growth of the water flea. In one experiment, he found that the time t, in days, required to reach a population of N is given by the relation e0.44t=NN0(228N0228N)4.46. Here N0 is the initial population size. If the initial population size is 50, how long is required for the population to grow to 125?arrow_forwardColonial Population The table shows the colonial population estimates of the American colonies for each decade from 1700 through 1780. Let y represent the population in the year t, with t=0 corresponding to 1700. (a) Use the regression feature of a graphing utility to find an exponential model for the data. (b) Use the regression feature of the graphing utility to find a quadratic model for the data. (c) Use the graphing utility to plot the data and the models from parts (a) and (b) in the same viewing window. (d) Which model is a better fit for the data? Would you use this model to predict the population of the United States in 2020? Explain your reasoning.arrow_forward
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