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Objective: To work with growth and decay models Words: The rate of change of a variable y is proportional to y and y(0) = y0 dy Differential Equation: dt ky k is the growth or decay constant General Solution: Uses: Biology, Chemistry, Economics, Earth Science, absorption of medication in the bloodstream, discharge and filtration of pollutants in rivers, and lots more. Ex 1. The ½ life of carbon 14 is 5730 years. If 100 grams of radioactive carbon 14 are stored in a cave for 1000 years, how many grams will be left at that time? What is the average amount of radioactive material in the cave for the 1000 years? What is the average rate of change for the radioactive carbon 14 over the 1000 years? Objective: To work with growth and decay models Ex 2. Newton’s Law of Cooling states that the rate of change in the temperature of an object is proportional to the difference between the object’s temperature and the temperature of the surrounding medium. Let y represent the temperature (in ˚F) of an object in a room whose temperature is kept at a constant 60˚. If the object cools from 100˚ to 90˚ in 10 minutes, how much longer will it take for its temperature to decrease to 80˚? Ex 3. Newton Second Law of Motion Force = mass x acceleration; We’ve been working with freefalling objects where air resistance has been ignored and the force was the gravitational pull. Now… a = dv/dt = gravity – air resistance and air resistance is proportional to the object’s velocity. a = dv/dt = g – k1v v is velocity and k1 is constant of proportionality, and g is a gravitational constant. F = ma F = m(dv/dt) = m(g – k1v) = mg – kv Which is F = Fg + Fr Ex 3. An object of mass m is dropped from a hovering helicopter. Find its velocity as a function of time t, assuming that the air resistance is proportional to the velocity of the object. Objective: To work with growth and decay models