Using Exponential and Logarithmic Functions

*7.8

*Example 1

a) The half-life of Sodium-22 is 2.6 years. Determine the value of k and the equation of decay for Sodium-22.

b) A geologist examining a meteorite estimates that it contains only about 10% as much Sodium-22 as it would have contained when it reached the surface of the Earth. How long ago did the meteorite reach the surface of the Earth?

*Example 2

*a) The half-life of radioactive iodine used in medical studies is 8 hours. What is the value of k for radioactive iodine?

*b) A doctor wants to know when the amount of radioactive iodine in a patient’s body is 20% of the original amount. When will this occur?

*Example 3

a. In 2007, the population of China was 1.32 billion. In 2000, it was 1.26 billion. Determine the value of k, China’s relative rate of growth.

b. When will China’s population reach 1.5 billion?

*Example 4

India’s population in 2007 was 1.13 billion and can be modeled by y = 1.13e0.015t. Determine when India’s population will surpass China’s.

(Note: t represents years after 2007.)

*7.8 Day 2 – Logistic Growth

These functions will have a horizontal asymptote at y = c

* A city’s population in millions is modeled by

where t is the number of years since 2000.

a) Graph the function.

b) What is the horizontal asymptote?

c) What will be the maximum population?

d) According to the function, when will the city’s population reach 1 million?

The horizontal asymptote is at f (t) = 1.432.

The population will reach a maximum of a little less than 1,432,000 people.

The graph indicates the population will reach 1 million people at t ≈ 3. Solving for t in the equation yields t = 2.78 years. So, the population of the city will reach 1 million people by 2003.