4.8 Exponential and Logarithmic Models

Exploring - 3 Exponential ModelsUninhibited Growth/Decay

Exponential growth/decay with limiting factors• Logistic Model

Newton’s Law of Cooling• Temperature decreases exponentially over time

Examples of growth/decay models

Examples which demonstrate the growth model:• Populations of people• Populations of organisms (cells, bacteria, …)

Examples which demonstrate the decay model:• Radioactive decay• Carbon 14 dating

The amount A of an object present at time t is given by

where A0 is the original amount of the object and k is the rate of decay or growth (k > 0 if growth and k < 0 if decay)

Exponential Growth and Decay

A(t) Aoekt

k > 0 k < 0

Uninhibited Growth ModelExample 1: A colony of bacteria grows according to the law of uninhibited growth. If 100 grams of bacteria are present initially, and 250 grams are present after two hours, how many will be present after 4 hours?

Uninhibited Decay Model (Half-Life)Example 2: The half-life of Uranium-234 is 200,000 years. If 50 grams of Uranium-234 are present now, how much will be present in 1000 years (Half-life is the time required for half of a radioactive substance to decay).

The temperature u of a heated object at time t is :

where T is constant temperature of surrounding medium, is the initial temperature of the object, and k < 0

Newton’s Law of Cooling

kteTuTtu )()( 0

0u

Example 3: A thermometer at 8 C is brought into a room with constant temperature 35 C. If the thermometer reads 15 C after 3 minutes, what will it read after 5 minutes?

In logistic models, the population P after time t obeys

where a, b, and c are constants with c > 0, and c represents the carrying capacity.

Logistic Growth and Decay Models

P(t) c

1 ae bt

b > 0b < 0

Logistic GrowthThe logistic growth model

represents the amount of bacteria (in grams) after t days

P(t) 500

1 6.67e 0.2476t

1) Graph the function using a graphing utility.

2) What is the carrying capacity?

3) What was the initial amount of bacteria?

Logistic GrowthThe logistic growth model

represents the amount of bacteria (in grams) after t days

P(t) 500

1 6.67e 0.2476t

4) When will the amount of bacteria be 300 grams?

4.8 Exponential and Logarithmic Models

Homeworkp. 334, #1, 3, 6, 7, 16, 23, 24e, 30d