6.4 Exponential Growth and Decay Law of Exponential Change Continuously Compounded Interest Radioactivity Newton’s Law of Cooling Resistance Proportional to Velocity

Law of Exponential Change If y changes at a rate proportional to the amount present ( ) and when t = 0, then Where k > 0 represents growth and k < 0 represents decay. The number k is the rate constant of the equation.

Developing the formula: Separating the variables, retain the k with the dt since k is only a constant, not a variable. Integrate each side, add the constant of integration on the independent side. Solve for y by using the exponential function on each side.

In a colony of fruit flies there are 100 flies at time 2 days and 300 flies at time 4 days. How many flies were there initially?

Radioactivity When an atom emits some of its mass as radiation, the remainder of the atom reforms to make an atom of some new element. This process of radiation and change is called radioactive decay,and an element whose atoms go spontaneously through this process is radioactive. The half-life of a radioactive element is the time required for half of the radioactive nuclei present in a sample to decay.

A radioactive element has a half-life of years. Initially there are 10 grams of this element present, how long will it take for the element to decay to 1 gram?

Find the exponential equation that goes through the two points: ( 0, 1.1 ) and ( -3, 3 ) where there are in the form ( t, y)

Newton’s Law of Cooling The rate of change in the temperature of an object is proportional to the difference in the object’s temperature and the temperature of the medium in which the object is placed. This gives us the following differential equation:

Solving this differential equation for T, we have: We’re dealing with initial temperature ( t = 0), temperature of the surrounding medium, and temperature at any time t.

An object is placed in an atmosphere of 60 degrees F. At the time it was placed there it’s temp was 100 degrees. Ten minutes later the temperature dropped to 90 degrees. How long will it take to drop to a temperature of 80 degrees?

Using data to work with Newton’s Law of Cooling Going back to the derivation of the formula, there was a point where the formula looked like this: When the calculator finds an exponential regression equation, it is of the form :

When finding the regression equation, use the ordered pairs Time (sec.)T (Celsius)T-T(s) (Cel) In the following problem, the surrounding temp is 4.5 degrees C

Using the data, the regression equation that calculator gives is: Remember that y is T – 4.5, so then our ‘model’ for T becomes: Looking at the form and comparing it to the original equation, we see that is the initial temperature of the object. The problem requires that we determine when the object will cool to 8 degrees Celsius. We’re going to use the calculator to solve the problem. Paste the regression equation into the y= and also insert in y = the equation y = 8. Solve for the point of intersection. We should get time = seconds.

Resistance Proportional to Velocity Resisting force is proportional to the objects’ velocity. Force = mass times acceleration Force == constant times velocity

Solving the differential equation, we get the following: Remembering that velocity is the derivative of position: Solving for position, (you can look at example 6 part b in your text), we get this :

As an object moves on indefinitely, the resisting force is causing it to slow down as time moves on. If time is allowed to go to infinity, we would find the total coasting distance of the object. We can look at this as a limit ! Coasting distance =

An ice skater of mass 50 kg has an initial velocity of 7 meters per second. If the k constant is 2.5 kg/sec, how long will it take her to coast from 7 m/sec to 1 m/sec? How far does she go before coming to a complete stop? Coasting distance is :