Using Exponential Functions to Measure and Compare Values

M12P B6 NOTES USING EXPONENTIAL FUNCTIONS TO MEASURE AND COMPARE VALUES

A. EXPONENTIAL FUNCTIONS OF TIME

1. POPULATION GROWTH example) The population of the planet Suluclac grows so that every 80 years, the population increases tenfold. a) Find a formula to find the b) How many times as large as c) How many times as large as population, A, of the planet after the present population will the the population 50 years from now t years, if the present population population be 50 years from will the population be 210 years now? from now? is A0 .

2. COMPOUND INTEREST example) The value of an investment increases threefold every 20 years. a) Find a formula to represent the value of the b) Find the ratio of the value of the investment after investment, A dollars, after t years if the present 135 years compared with the value of the investment after 55 years. value of the investment is A0 dollars.

tt In an exponential function of time, T , where A  A0(b) _____= elapsed time If the value after time t1 is A1 , and the value after _____= multiplying factor time t2 is A2 , then the ratio of these values is:

_____= multiplying time period A 2  _____= starting value A1

_____= value after elapsed time t

When you find the ratio of the values of an exponential function of time measured at two different times, do you need to know the starting value?

Explain why. OTHER EXAMPLES OF COMPARING VALUES USING EXPONENTIAL FUNCTIONS

4. EARTHQUAKES

The intensity of an earthquake is measured by the amount of ground motion it causes. Because the intensities vary from very small to very large, to compare earthquakes, their intensities are written as an exponential function of the Richter Scale magnitude, where:

Each increase of 1 unit in magnitude on the Richter Scale represents a 10-fold increase in intensity.

Use the above description of the Richter Scale to write a formula to find the intensity, I, of an earthquake of magnitude R, if the intensity of a magnitude 0 earthquake is I0 .

Eg) How many times as intense as an earthquake of Eg) The earthquake in Colombia in 1983 had a magnitude 2.4 is an earthquake of magnitude 6.7? magnitude of 5.5. The earthquake in Ladner in 1 2000 was 40 as intense as the Colombia earthquake in 1983. Find the Richter Scale magnitude of the the Ladner 2000 earthquake.

5. LOUDNESS OF SOUNDS

The loudness of sounds is described in decibels (dB), where Every increase of 10 dB represents a tenfold increase in loudness. a) If the loudness of a 0 dB sound (the threshold of hearing) is L0 , find the formula to describe the loudness of a sound, L, with a measurement of x decibels.

b) Compare, using a ratio, the loudness of a sound from a Jet engine at 25 m (140 dB) to the loudness of a conversation (60 dB)

When comparing the loudness of different sounds using ratios, the constant ______cancels. c) Find the ratio of the loudness of a sound of x d) Use the formula from c) and the Decibel Scale decibels to a sound of y decibels. from p. 100 of Addison-Wesley to calculate how many times as loud as a refrigerator hum is a rock group. 6. ACIDITY AND ALKALINITY OF SOLUTIONS

The pH scale measures the acidity or alkalinity of a solution, where

Each 1-unit decrease in pH represents a tenfold increase in acidity. Each 1-unit increase in pH represent a tenfold increase in alkalinity.

If the acidity of a pH 0 solution is A0 , find a formula to calculate the acidity of a solution with pH x.

eg) Compare, using a ratio, the acidity of tomato eg) Use the pH scale from p. 101 or Addison- juice (pH 4) to the acidity of baking soda (pH 9) Wesley to calculate how many times as acidic as pure water is lemon juice.

eg) If the alkalinity of a pH 0 solution is Al0 , find a formula to calculate the alkalinity of a solution with pH x.

eg) Use the formula from e) and the pH scale from p. 101 to calculate how many times as alkaline as baking soda is oven cleaner.