College Algebra Class Notes – Lesson 4.8 Exponential and Logarithmic Models

Section 4.8

1. Exponential Growth and Decay ���� = ����

� = the original amount of the material at t = 0

� =growth/decay rate

Example 1: Exponential Growth. 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?

Definition: Half-life is the time required for half of the radioactive substance to decay. Half-life is the amount of time

required for the amount � to decay to

��.

Example 2: Exponential Decay. A radioactive material, Strontium 90, decays according to ��� = 500��.�����, where

A(t) is the amount present at time t (in years).

a) What is the decay rate?

b) When will 400 grams of the material be left?

c) What is the half-life?

Example 3: Half-Life Problem (Exponential Decay). 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).

��� = ����

Growth Model

where k > 0

��� = ����

Decay Model

where k < 0

College Algebra Class Notes – Lesson 4.8 Exponential and Logarithmic Models

2. Newton’s Law of Cooling.

Example 3: A thermometer at 8°� is brought into a room with

constant temperature 35°�. If the thermometer reads 15°�

after 3 minutes, what will it read after 5 minutes?

3. Logistic Model

y = c y = c

Carrying capacity, c, is the value the population approaches, as t approaches infinity.

Example 4: Suppose we are given a logistic growth model ���� =500

1+6.67�−.2476� which represents the amount of

bacteria (in grams) after t days.

a) Graph the function using a graphing utility

b) What is the carrying capacity?

c) What was the initial amount of bacteria?

d) When will there be 300 grams of bacteria?

Newton’s Law of Cooling

$��� = % + �$� − %����

T = constant temperature of surrounding medium

$� = initial temperature of the heated object

k = rate of cooling (Note: k < 0)

&��� ='

1 + (��)�

Logistic Growth Model

where a, b, and c are constants.

c > 0 and b > 0

&��� ='

1 + (��)�

Logistic Decay Model

where a, b, and c are constants.

c > 0 and b < 0