Warm Up Dec. 19 Write and solve the differential equation that models the verbal statement. Evaluate the solution at the specified value. The rate of change of B is proportional to B. When B = 2, t = 0 and when B = 8, t = 3. What is the value of t when B = 10?

Write the exponential function y = y 0 e k t whose graph passes through the two points (0, 3) and (2, 10) Write the exponential function y = y 0 e k t whose graph passes through the two points (1, 1.5) and (3, 6)

Exponential Growth or Decay Any situation where the rate of change of a function is proportional to the value of the function… For example, any kind of population growth, radioactive decay, etc.

the rate of change of a function, P, is proportional to the value of the function, P,…in other words Use separation of variables to solve…

In 1980 the population of a town was 21,000 and in 1990 was 20,000. If the population is following an exponential pattern of decline, what is the expected population in 2100? (Let t = 0 be 1980)

Example Suppose a colony of bacteria is growing (exponentially) in the corner of your shower stall. On June 1, there are 1 million bacteria there. By July 1, there are 7.5 million. Your shower stall can contain a total of 1 billion bacteria. When will you have to start takings showers down at the gym?

The half life of radioactive carbon-14 is 5,750 years. There are 10 grams present initially. Write an equation in the form y = Ce kt to represent the amount present at anytime t. How much will remain of the 10 grams after 3000 years? How long will it take for the 10 grams to be reduced to 1 gram?

The half life of radioactive carbon-14 is 1,250 years. There are 10 grams present after 500 years. What was the initial quantity of the substance? How much will remain after 3000 years?

Write and solve the differential equation that models the verbal statement. The rate of change of y with respect to t is proportional to y – 70.

Hiram Fentley, heir to the Fentley Feta Cheese fortune was found dead at his home at 2:30 am. The temperature in the room was a constant 70 degrees. His body temperature at 3:00am was 85 degrees and at 4:00am was 78 degrees. At what time was he killed? 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. Which when solved is… T= body temperature R = temp of the surroundings t = time

Let P(t) represent the number of wolves in a population at time t years, when t > 0. The population P(t) is increasing at a rate directly proportional to 800 – P(t), where the constant of proportionality is k. 1)Write a differential equation that models this growth. 2) If P(0) = 500 and P(2) = 700, write an equation P that models the given situation. 3) Find