4.5 Modeling with Exponential and Logarithmic Functions

Exponential Growth Model A population experiences exponential growth increases according to the model Where n(t)=population at time t n 0 =initial population size r=relative rate of growth t=time

Predicting the Size of a Population The initial bacterium count in a culture is 500. A biologist later makes a sample count of bacteria in the culture and finds that the relative rate of growth is 40% per hour. – Find a function that models the number of bacteria after t hours. – What is the estimated count after 10 hours? – Sketch the graph of the function

Radioactive Decay If m 0 is the initial mass of a radioactive substance with half-life h, then the mass remaining at time t is modeled by the function where

Radioactive Decay Polonium-210 has a half-life of 140 days. Suppose a sample of this substance has a mass of 300 mg. – Find a function that models the amount of the sample remaining at time t. – Find the mass remaining after one year. – How long will it take for the sample to decay to a mass of 200 mg? – Draw a graph of the sample mass as a function of time.

Newton’s Law of Cooling If D 0 is the initial temperature difference between an object and its surroundings, and if its surroundings have temperature T s, then the temperature of the object at time t is modeled by the function where k is a positive constant that depends on the type of object.

Newton’s Law of Cooling A cup of coffee has a temperature of 200 o F and is placed in a room that has a temperature of 70 o F. After 10 min the temperature of the coffee is 150 o F. – Find a function that models the temperature of the coffee at time t. – Find the temperature of the coffee after 15 min. – When will the coffee have cooled to 100 o F?

Logarithmic Scales The pH scale The Richter scale The Decibel scale

pH Scale and Hydrogen Ion Concentration The hydrogen ion concentration of a sample of human blood was measured to be. Find the pH. The most acidic rainfall ever measured occurred in Scotland in 1974; its pH was 2.4. Find the hydrogen ion concentration.

Magnitude of Earthquakes In 1906 earthquake in San Francisco had an estimated magnitude of 8.3 on the Richter Scale. In the same year a powerful earthquake occurred on the Colombia- Ecuador border and was four times as intense. What was the magnitude of the Colombia- Ecuador earthquake on the Richter Scale?

Intensity of Earthquakes The 1989 Loma Prieta earthquake that shook San Francisco had a magnitude of 7.1 on the Richter Scale. How many times more intense was the 1906 earthquake than the 1989 event?

Sound Intensity of a Jet Takeoff Find the decibel intensity level of a jet engine during takeoff if the intensity was measured at 100W/m 2.