Slide 8 Def

• Work out Warm-ups• Find a radioactive decay problem

• Carbon Dating and Depreciation Worksheet.

Logarithms and Exponentials-Depreciation of Value

Objectives: 1.) Review the concept of exponentials and

logarithms as applicable to compound interest and continuously compounded interest.

2.) Apply the concept of exponentials and logarithms to depletion of value problems and carbon dating

problems

Recall the three Interest Equations

trIP 1

nt

n

rIP

1 Compound Interest

Basic Interest

rtIeP Continuously Compounded Interest

• For rates on CD’s and savings accounts, with them being so low in rates the best rates we have are 1.25 on savings and 2.5 on a CD I do believe!! Loans range from 4.25-9% mostly but we have some that can go 18% on our not so good customers.

Warm- ups 10 minutes

1.) In your own words, describe the difference between compound interest problems and continuously compounded interest problems.

2.) Approximately what is the value of e? 3.) How long will it take for $1000 to grow to $1500 if it earns 2.5%

interest, compounded monthly? 4.) How long will it take for a $2000 investment to double if it earns a

2.5% inter rate compounded continuously? 5.) How much money will you obtain after 40 years if you invested

$5000 in an account that collects 2.5% interest compounded continuously?

Let us look at Continuously Compounded Interest Rates in Nature

• Population Growth and Carbon Dating

• Recall, we studied the idea of taking an investment amount and continuously compounding a rate set at 100%.

– P= I(1 + )nt n1 nas

We came up with

P = Iert

I= Initial Investment AmountP= Final Amount/payoutr = annual interest ratet= time in years

Population Growth

• P= aekt

P is the population a= a fixed amountk= rate of growtht= time

• Example 1 and 2 on pages237 and 238

• Page 244 and 245• #35; 37

Carbon Dating/ Radioactive Decay

• Radioactive Decay: Radioactive decay involves the spontaneous transformation of one element into another

• Half Life: The time required for half the nuclei in a sample of a specific isotopic species to undergo radioactive decay.

• Carbon Dating: Carbon-14 dating is a way of determining the age of certain archeological artifacts of a biological origin up to about 50,000 years old by looking at the radioactive carbon-14 content

• http://serc.carleton.edu/quantskills/methods/quantlit/RadDecay.html

Carbon Dating/ Radioactive Decay• Page 239 “Study Tip”

Radioactive Decay equation: y = ae-bt

• y is the amount element you have left• a= initial amount• e is e• b is the rate of deterioration• t is the time the element has deteriorated for.

Looking at Radium226

For example: Radium226 is a radioactive element. Its half life is 1620 years.

Suppose you have 10 grams of it.

)1620(21 1010 re

1.) Use the half life to find the rate in which it deteriorates. Find “r”2.) Then find any y amount with respect to time.

Ho much radioactive radium226 will remain after …

a.) 500 years?

b.) 3240 years?Ian is cool

Page 244 #25

Homework: page 244 & 245 #25-30;

Vocabulary- Depreciation

• Defined as the permanent decrease in the quality, quantity or the value of an asset over time

• Computers, cars, production machinery,…

Suppose you bought a computer

• Computers generally depreciate in value about 20% each year.

• The Macbook Pro with the 17” screen and all the bells and whistles is sold for $2,300 at the Mac store ( ridiculously unfortunate). If you bought this Mac, how much is its resale value in 1 year? 2 years? 3 year?

$2,300

1 year 2 years 3 years

$2,300

1 year 2 years 3 years

Initial ValueThe question: The value in … ?

20 % Depreciation Rate Per Year

Initial Value: $ 2,300Question: The value in 1 year, 2 years, 3 years…, t years?20% Depreciation rate per year

In 1 year:

In 2 years:

In 3 years:

Depreciation Equation

P = I(1 – r)t

P =Depreciated ValueI = Original amount r= depreciation ratet= time in years

Cars generally lose 15% to 20% of their value a year.

• A BMW Z4 2000 had a retail price of $46,500.00. Every year, the car depreciates in value 15%. If you were to buy it this year, 2011, approximately how much would you expect to buy it for?

• John Bros. acquired a large production print machine on 1st Jan 2009 at a cost of $14,000. The firm writes off the depreciation at 20% every year.