Uninhibited Growth Formula (Cells, Bacteria, Etc

Math Analysis: Chapter 7-Exponential and Logarithmic Functions Name: Notes on Section 7-6: Growth and Decay (Page 1)

kt Uninhibited Growth Formula (cells, bacteria, etc.): N(t)  N0e k  0

Where N0 stands for the initial amount of the substance, N(t) is the amount after t units of time, and k is a constant value associated with the substance.

Example 1: A culture of bacteria grows at an exponential rate. If 100 grams are present initially and there is 175 grams after 1 hour, how much would be present after 4 hours?

Example 2: For a certain strain of bacteria, k = .0325 and t is measured in days. How long will it take 25 bacteria to increase to 500?

kt Uninhibited Radioactive Decay Formula: A(t)  A0e k  0

Where A0 stands for the initial amount of the substance, A(t) is the amount after t units of time, and k is a constant value associated with the substance.

Example 3: 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. NOTE: The half-life is the time required for half of the radioactive substance to decay.

Example 4: If 18 grams of a certain radioactive substance decays to 5 grams after 340 years, what is its half-life?

Example 5: The half-life of a certain radioactive substance is 2500 years. If 35 grams of the substance are present now, how much will be present in 1500 years? Math Analysis: Chapter 7-Exponential and Logarithmic Functions Name: Notes on Section 7-6: Growth and Decay (Page 2)

u(t)  T  (u T )ekt k  0 Newton’s Law of Cooling Formula: 0

Where T = temperature of surrounding medium, u0 is the initial temperature of the heated object, and u(t) is the temperature of the object after t minutes, and k is a constant associated with the substance that is cooling

Example 6: A cup of hot chocolate is 100 degrees Celsius. It is allowed to cool in a room whose air temperature is 22 degrees Celsius. If the temperature of the hot chocolate is 85 degrees Celsius after 4 minutes, when will its temperature be 60 degrees Celsius?

Example 7: Find the time it takes a cup of soup to cool to 100 degrees, if it has been heated to 180 degrees. Room temperature is 70 degrees and k = .01.

Exponential Growth Formula (for populations, how much a house is worth, etc.): t N(t)  N0 1 r

Where N0 stands for the initial amount of the population, N(t) is the population amount after t years, and r is the rate of increase of the population.

Example 8: If a population grows at a rate of 2.5% per year, and is 60,000 now, what will be the size of the population in 5 years? In 8 years?

Example 9: A certain house appreciates (grows) in value exponentially at a rate of 2.7% per year, and is worth $375,000 now. What will the house be worth 3 years from now?

Example 10: A certain rare coin, which is valued at $10, appreciates at a rate of 1.5% per year. What will the coin be worth 10 years from now? Math Analysis: Chapter 7-Exponential and Logarithmic Functions Name: Notes on Section 7-6: Growth and Decay (Page 3)

t Exponential Depreciation Formula (for cars, boats, etc. that depreciate in value): N(t)  N0 1 r

Where N0 stands for the initial amount of the substance, N(t) is the amount after t years, and r is the rate of decrease (or decay).

Example 11: A car the costs $17,500 depreciates at a rate of 16.5%. How much will it be worth in 4 years? In 7 years?

Example 12: The value of a $38,000 yacht decreases at a rate of 12% per year. Find its value after 5 years.

Now, try the problems below (you will have to figure out which formula to use with each problem)

Problem 1: If 100 kilograms of a certain radioactive substance decays to 32 kilograms after 6500 years, what is its half-life?

Problem 2: A certain motorcycle is worth $27,000 currently, but is depreciating at a rate of 15% annually. How much will the motorcycle be worth 6 years from now?

Problem 3: A cup of tea is heated to 125 degrees Celsius. It is allowed to cool in a room whose air temperature is 30 degrees Celsius. If the tea cools to 98 degrees Celsius after 6 minutes, when will its temperature be 72 degrees Celsius?

Problem 4: In the city of Fayetteville, the population grows at a rate of 3.6% per year, and is 38,000 now. What will be the size of the population 8 years from now?

Problem 5: A certain bacteria grows from 5 grams to 39 grams after 2 and ½ hours. How much of this bacteria will be present after 9 hours?