MAT 171 Precalculus AlgebraTrigsted - Pilot Test

Dr. Claude Moore - Cape Fear Community College

CHAPTER 5: Exponential and Logarithmic

Functions and Equations

5.1 Exponential Functions5.2 The Natural Exponential Function5.3 Logarithmic Functions5.4 Properties of Logarithms5.5 Exponential and Logarithmic Equations 5.6 Applications of Exponential and Logarithmic Functions

·Solving Compound Interest Problems·Exponential Growth and Decay·Solving Logistic Growth Applications·Using Newton’s Law of Cooling

Objectives

Periodic Compound Interest Formula

A = Total amount after t yearsP = Principal (original investment)r = Interest rate per yearn = Number of times interest is compounded per yeart = number of years.

Find the Doubling Time: To find the time it will take for an investment to double A becomes 2P.

How long will it taken (in years and months) for an investment to double if it earns 6.4% compounded quarterly.

A = Total amount after t years = 2PP = Principal =Pr = Interest rate per year = 6.4% = .064n = Number of times interest is compounded per year = 4t = number of years

Suppose an investment of $7500 compounded continuously grew to an amount of $8320 in 18 months. Find the interest rate and then determine how long it will take for the investment to grow to $10000.

A = Total amount after t years = $8320P = Principal= $7500r = interest rate per yeart = number of years= 1.5

A = Total amount after t years = $10000P = Principal= $7500r = interest rate per year = .069t = number of years

Exponential Growth

A model that describes the exponential uninhibited growth of a population, P, after a certain time, t, is

P(t) = P0ekt

Where P0 = P(0) is the initial population and k > 0 is a constant called the relative growth rate. ( Note: k is sometimes given as a percent.)

The population of a small town grows at a rate proportional to its current size. In 1925, the population was 15000. In 1950, the population had grown to 25000. What was the population of this town in 1970? Round to the nearest whole number.

Half Life

The required time for a given quantity of an element to decay to half of its original mass.

Suppose that an item is found containing 7% of its original amount of carbon. If the half life of that element is 25 years, how old is the item?

A0 is the original amount of carbon.A(25)= ½ A0 because the half life of the item is 25 years.

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Suppose that an item if found containing 7% of its original amount of carbon. If the half life of that element is 25 years, how old is the item?

The item is about 96 years old.

Logistic ModelDescribes population growth when outside limiting factors that affect population growth exist.

y = C

Logistic GrowthA model that describes the logistic growth of a population P at any time t is given by the function

Where B,C and k are constants with C > 0 and k < 0.

Six fish were introduced into a small backyard pond. Because of limited food, space, and oxygen, the carrying capacity of the pond is 100 fish. The fish population at any time t, in days is modeled by the logistic growth function.

If 10 fish were in the pond after 15 days, ·Find B·Find k·How many fish will be in the tank after 60 days? Round to the nearest whole number.

·F(0) = 6 Initial amount of fish is 6 C = 100 Capacity of the pond is 100

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Six fish were introduced into a small backyard pond. Because of limited food, space, and oxygen, the carrying capacity of the pond is 100 fish. The fish population at any time t, in days is modeled by the logistic growth function.

If 10 fish were in the pond after 15 days, ·B = 47/3·Find k·When will the pond contain 60 fish? Round to the nearest whole number.

b. F(15) = 10

Continued on the next screen

Six fish were introduced into a small backyard pond. Because of limited food, space, and oxygen, the carrying capacity of the pond is 100 fish. The fish population at any time t, in days is modeled by the logistic growth function.

If 10 fish were in the pond after 15 days, ·B = 47/3·Find k

c. When will the pond contain 60 fish? Round to the nearest whole number.

c. F(60)