1. [-/6 Points] DETAILS MY NOTES A certain drug is administered intravenously to a patient at the continuous rate of 5 mg per hour. The patient's body removes the drug from the body at a rate that is proportional to the amount of drug in the blood. Write the amount of drug in the blood as y and let the positive constant of proportionality for the removal rate be k. Write a differential equation that is satisfied by the amount of drug in the blood. dy dt Submit Answer 2. [-/6 Points] DETAILS MY NOTES A porous material dries outdoors at a rate that is proportional to the moisture content. Set up the differential equation whose solution is y(t), the amount of water at time t in a towel on a clothesline. (Note: Use k for the constant of proportionality with k > 0.) , k >o Submit Answer 3. [-/6 Points] DETAILS MY NOTES An experimenter reports that a certain strain of bacteria grows at a rate proportional to the square of the size of the population. Set up a differential equation that describes the growth of the population. (Note: Use k for the constant of proportionality with k > 0.) , k >o Submit Answer 4. [-/6 Points] DETAILS MY NOTES L.F. Richardson proposed a model to describe the spread of war fever. + His proposal: If y(t) is the percent of the population advocating war at time t, then the rate of change of y(t) at any time is proportional to the product of the percentage of the population advocating war and the percentage not advocating war. Set up a differential equation that is satisfied by y(t). (Note: Use k for the constant of proportionality with k > 0.) , k >o Submit Answer 5. [-/14 Points] DETAILS MY NOTES A student must learn M unfamiliar words for an upcoming test. The rate at which the student learns is proportional to the number of items remaining to be learned, with constant of proportionality equal to k. Initially, the student knows none of the words. Let y(t) stand for the number of the words that the student knows at time t. (a) Write down the right hand side of the differential equation satisfied by y. (Your answer should be given in terms of y.) dy (b) What is the initial value of y? (Give an exact answer.) y(0) = (c) Suppose that there are 120 words to be learned and that k = 0.7 per hour for this student. Recall that the solution to y' = k(M - y) with y(0) = 0 is given by y(t) = M(1 - e-kt). (Give your answers correct to at least three decimal places.) How many hours would it take the student to learn the first 30 percent of the words? hours How long would it take to learn the next 30 percent of the words? hours How long would it take to learn the next 30 percent of the words? hours (d) The student has just finished learning the first 30 percent of the words. (Give your answers correct to at least three decimal places.) How long would it take to learn 30 percent of the remaining words? hours After doing this, how long would it take to learn 30 percent of the words that then remain? hours Submit Answer 6. [-/15 Points] DETAILS MY NOTES A certain hormone is produced by an endocrine gland, causing its concentration to increase at a constant rate of A mg/L/hr. The hormone is metabolized by the liver, with the rate of elimination being proportional to the concentration of the hormone. The constant of proportionality in this relationship is given by the (positive) constant k. The concentration of the hormone at time t is written as y(t) (a) Write down the right side of the differential equation satisfied by y. (Your answer should be in terms of y.) dy (b) This differential equation can be rewritten in the more familiar form - = k(M - y) for a particular choice of M. Write down an expression for M in terms of A and k. M = (c) If the initial concentration is zero, the solution of the differential equation is y(t) = M(1 - ekt). In terms of A and k, what value does the hormone concentration approach in the long run? mg/L At time t = 0, a patient comes off a drug treatment that had blocked production of the hormone. (So, at that time, they do not have any of the hormone in their body.) Their doctor finds that the initial rate of increase of their hormone concentration is 0.3 mg/L/hr and that, in the long run, the concentration approaches 3 mg/L. (Give your answers correct to at least three decimal places.) d) After how long would the patient's hormone concentration have reached half of its long-term level? hours (e) At what rate would the level of hormone have been increasing at that time? mg/L/hr