Applications of exponential and logarithmic functions

1
Applications of exponential and logarithmic
functions

• from sections 11.3, 11.6, 11.7

2
Chapter 11 Notes

• Exponential growth
• Newtons Law of cooling
• Half-life of materials
• Appreciation/depreciation

3
Chapter 11 Notes

• Exponential growth
• y final amount
• n initial amount
• k constant
• t time in years

4
Chapter 11 Notes

• Exponential growth example
• The constant k for a radioactive substance is
  -0.08042. In how many years will a 250 gm
  sample reduce to 50 gm?

5
Chapter 11 Notes

• Newtons Law of cooling
• y temperature of object
• t time in minutes
• C ambient temperature
• a k constants

6
Chapter 11 Notes

• Newtons Law of cooling example
• An object is heated to 70F then taken outdoors
  where the temperature is 15F. What is the
  temperature of the object 30 minutes later if a
  55 and k 0.05?

7
Chapter 11 Notes

• Half-life of materials
• y final amount
• yo initial amount
• C constant of proportionality
• t time
• T time per C (half-life)

8
Chapter 11 Notes

• Half-life of materials example
• After 10 years, 5.1 kg of a radioactive material
  remains from a 14.3 kg sample. What is the
  half-life of the material?

9
Chapter 11 Notes

• Appreciation/depreciation
• A appreciated/depreciated value
• P present value
• r rate of appreciation () or depreciation (-)
• t time in years

10
Chapter 11 Notes

• Appreciation/depreciation example
• A couple buys a starter home for 52,000.
  They want to have 20,000 to make a down payment
  on a bigger house. If the current home
  appreciates at a rate of 5.5, how long do they
  need to wait to sell it to have the down payment
  for a newer home?

11
Chapter 11 Notes

• Problem-solving strategy
• Read through the problem using variables and
  wording to determine the type of application
  identify the appropriate formula.
• Substitute values into the formula and solve.

12
Chapter 11 Notes

• Examples
• A piece of office equipment valued at 25,000
  depreciates at a rate of 10 annually. In how
  many years will the equipment be worth 5000?
• A human body is found in a room where the
  temperature is 70F. The body temperature is
  measured to be 82F, how long ago did the person
  die? Use a 36.6 and k 0.01.

13
Chapter 11 Notes

• Example answers
• Use appreciation/depreciation formula
• A 5000, P 25,000, r -0.1
• solve for t about 15.3 years
• Use Newtons law of cooling formula
• y 82.4, a 36.6, k 0.01, C 70
• solve for t 108 minutes (1 hr., 48 min.)