Applications of Exponential Functions. Radioactive Decay Radioactive Decay The amount A of radioactive material present at time t is given by Where A.

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Presentation transcript:

Applications of Exponential Functions

Radioactive Decay Radioactive Decay The amount A of radioactive material present at time t is given by Where A 0 is the initial amount at t=0 and h is the material’s half-life.

Example 1: The half-life of radium is approximately 1600 years. How much of a 1-gram sample will remain after 1000 years? The half-life of radium is approximately 1600 years. How much of a 1-gram sample will remain after 1000 years?

Example 1 Solution: 0.65 g remains after 1000 yrs.

Oceanography The intensity I of light (in lumens) at a distance x meters below the surface of a body of water decreases exponentially by: The intensity I of light (in lumens) at a distance x meters below the surface of a body of water decreases exponentially by: where I 0 is the intensity of light above the water.

Example 2: For a certain area of the Atlantic Ocean, I 0 =12 and k=0.6. Find the intensity of light at a depth of 5 meters in this body of water.

Example 2: Given I 0 =12 and k=0.6 and x=5 : lumens

Malthusian Population Growth Malthusian model for Population Growth assumes a constant birth rate (b) and death rate (d). It is as follows: Malthusian model for Population Growth assumes a constant birth rate (b) and death rate (d). It is as follows: where k=b - d, t is time in years, P is current population, and P 0 the initial population. where k=b - d, t is time in years, P is current population, and P 0 the initial population.

Example 3: The population of the U.S. is approximately 300 million people. Assuming the annual birth rate is 19 per 1000 and the annual death rate is 7 per What does the Malthusian model predict the population will be in 50 years? The population of the U.S. is approximately 300 million people. Assuming the annual birth rate is 19 per 1000 and the annual death rate is 7 per What does the Malthusian model predict the population will be in 50 years?

Example 3: Given: b=0.019, d=0.007, Given: b=0.019, d=0.007, P 0 =300 million, t=50 P 0 =300 million, t=50 Prediction in 50 years

Epidemiology

Example 4: In a city with a population of 1,200,000, there are currently 1,000 cases of infection with HIV. In a city with a population of 1,200,000, there are currently 1,000 cases of infection with HIV. Using the formula: Using the formula: How many people will be infected in 3 years? How many people will be infected in 3 years?

Example 4: Infected in 3 years

Example 5:

Using our graphing calculator, the approximate intersection of the two functions at (71,4160) gives us the prediction: In about 71 yrs the food supply will be outstripped by population of about 4160.