Let C(t) be the concentration of a drug in the bloodstream. As the body eliminates the drug, C(t) decreases as a rate that is proportional to the amount of the drug that is present at a time. Thus, C'(t) = kC(t). a. If C0 is the concentration at time t=0, find the concentration at time t. b. If the body eliminates half the drug in 30 hours, how long does it take to eliminate 90% of the drug.

A culture of yeast grows at a rate proportional to its size. If the initial population is 8000 cells and it doubles after 3 hours, answer the following questions. 1. Write an expression for the number of yeast cells after t hours. Answer: P(t) = 2. Find the number of yeast cells after 5 hours. Answer: 3. Find the rate at which the population of yeast cells is increasing at 5 hours. Answer (in cells per hour):

If V(t) is the volume of water in a reservoir at time t (seconds), v(t) is the rate of change,  then  means

Newton’s law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e., the temperature of its surroundings). We let (t) denote time, T(t) be the temperature of the object at time (t), (T0 = T(0)) (the object’s temperature at (t = 0)), and Tomg be the constant ambient temperature. Set u(t) = T(t) - Tomg and show from Newton’s law of cooling that (u) satisfies the initial value problem (*) u’(t) = ku(t), u(0) = T0 - Tomg       for a constant (k). Solve the initial value problem (*), i.e., find u(t)) (expressed in terms of (k), T0, and Tomg . Find T(t).