6.2 Growth and Decay Law of Exponential Growth and Decay C = initial value k = constant of proportionality if k > 0, exponential growth occurs if k < 0, exponential decay occurs

Solving a Differential Equation Multiply both sides by y. Integrate both sides with respect to x. dy = y’dx because dy/dx = y’ Multiply both sides by 2. C = 2C

Exponential Growth and Decay Model If y is a differential function of t such that y >0 and y’ = ky, for some constant k, then Y = Ce kt C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0.

Proof: y’ = ky So, all solutions of y’ = ky are in the form of y = Ce kt

Ex. The rate of change of y is proportional to y. When t = 0, y = 2. When t = 2, y = 4. What is the value of y when t = 3? Because y’ = ky we know that y = Ce kt. We can find the values of C and k by applying the initial conditions. 2 = Ce 0 C = 2 4 = 2e 2k So, the model is When t = 3, the value of y is 2e (3) = 5.657

Newton’s Law of Cooling Let y represent the temperature (in o F) of an object in a room whose temperature is kept at a constant 60 o. If the object cools from 100 o to 90 o in 10 minutes, how much longer will it take for its temperature to decrease to 80 o ? From Newton’s Law of Cooling, the rate of change in y is proportional to the difference between y and 60. Separate variables first.

Take e to both sides. Using y = 100 when t = 0, solve for C. 40 = C Since y = 90 when t = 10 k = The model is: Finally, when y = 80, you obtain So, it will require more minutes for the object to cool to 80 o.

Ex. 2 Money is deposited in an account for which interest is compounded continuously. If the balance doubles in 6 years, what is the annual percentage rate? A = Pe rt 2P = Pe rt 2 = e 6r ln 2 = 6r

Ex. 1 A sample contains 1 gram of radium. How much radium will remain after 1000 years? (Use a half-life of 1620 years.) First we need to find k, the constant of proportionality. Take the ln of both sides.

Ex.Suppose that 10 grams of the plutonium isotope Pu-239 was released in the Chernobyl nuclear accident. How long will it take for the 10 grams to decay to 1 gram? Pu-239 has a half life of 24,100 years y = Ce kt We know that C = 10 grams at time t = 0. First, find k. 5 = 10e k(24,100) So, the model is y = 10e t To find the time it would take for 10 grams to decay to 1 gram, solve for t in 1 = 10e t = 80,059 years