8.8 – Using Exponential and Logarithmic Functions Ex 1) The intensity of sunlight reaching points below the surface of the ocean varies exponentially with the depth of the point below the surface of the water. Suppose that when the intensity at the surface is 1000 units, the intensity at the depth of 2 meters is 60 units. a) Find the exponential function.
b) What is the intensity at 10 meters?
c) Plants cannot grow beneath the surface if the intensity of sunlight is below 0.001 units. What is the maximum depth at which plants will grow? Ex 2) You accidentally inhaled some mildly poisonous fumes. Twenty hours later, you see a doctor. From a blood sample, she measures a poison concentration of 0.00372 milligrams per cubic centimeter (mg/cc), and tells you to come back in 8 hours. On the second visit, she measures a concentration of 0.00219 mg/cc. Let t represent the number of hours that elapsed since your first visit to the doctor and let C be the concentration of poison in your blood in mg/cc. Assuming that C varies exponentially with time: a) Find the exponential function.
b) You can resume normal activities when the poison concentration drops to 0.00010 mg/cc. How long after you inhaled the fumes will you be able to resume normal activities?
d) The biological half-life of this poison is length of time it takes for the concentration to drop to half of its value. What is the half- life of this poison? 8.8 Day 1 – Using Exponential and Logarithmic Functions
Exponential Decay: The depreciation of the value of a car is an example of exponential decay. When a quantity decreases by a fixed percent each year, or other period of time, the amount y of that quantity after t years is given by:
where a is the initial amount and r is the rate of decay.
Ex 1) A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for half of this caffeine to be eliminated from a person’s body?
Exponential Decay: Another model for exponential decay is given by:
where k is a constant.
* This model is used by scientists to solve problems involving radioactive decay.
Ex 2) The half-life of a radioactive substance is the time it takes for half the atoms of the substance to become disintegrated. All life on Earth contains the radioactive element Carbon-14, which decays continuously at a fixed rate. The half-life of Carbon-14 is 5760 years. That is, every 5760 years half of a mass of Carbon-14 decays away. a) What is the value of k for Carbon-14?
b) A paleontologist examining the bones of a woolly mammoth estimates that they contain only 3% as much Carbon-14 as they would have contained when the animal was alive. How long ago did the mammoth die? Exponential Growth: When a quantity increases by a fixed percent each time period, the amount y of that quantity after t time periods is given by:
where a is the initial amount and r is the rate of growth. Ex 3) In 1910, the population of a city was 120,000. Since then, the population has increased by exactly 1.5% per year. If the population continues to grow at this rate, what will the population be in 2010? Exponential Growth: Another model for exponential growth, preferred by scientists, is given by:
where k is a constant.
Ex 4) In 2007, the population of Georgia was 9.36 million people. In 2000, it was 8.18 million. a. Determine the value of k, Georgia’s relative rate of growth.
b. When will Georgia’s population reach 12 million people?
c. Michigan’s population in 2000 was 9.9 million and can be modeled by . Determine when Georgia’s population will surpass Michigan’s. Find Equations of Exponential Functions
Ex 5) Find an equation of an exponential function that passes through (2, 36) and (1, 12) with an asymptote at y = 0.
Ex 6) Find an equation of an exponential function that passes through (4, 26) and (6, 8) with an asymptote at y = 5.