MATH133 Unit 5: Exponential And Logarithmic Functions

MATH133                                                                               NAME:__________

Individual Project Assignment

 

 

Problem1 Photic Zone

 

Light entering waterinapond, lake, sea, or oceanwillbe absorbed or scattered bythe particles in thewater and its intensity willbe attenuated bythe depth of thewater, x, in feet. Marine lifein theseponds, lakes,seas, andoceans dependonmicroscopic plant lifethat exists inthe photiczone. Thephoticzoneis from thesurfaceofthe water down to a depth in that particular bodyofwaterwhereonly1% of thesurfacelightremainsunabsorbedornotscattered. The equation that models this light intensity is:

In this exponential function,  is theintensityof thelight at the surfaceof thewater, k is a constant based on theabsorbingor scatteringmaterials in that bodyof waterand is usuallycalled the coefficient ofextinction, e is thenatural number e ≅2.718282, andis thelight intensityat x feet below thesurfaceofthe water.

a.  Chooseavalueof kbetween0.025and0.095.

b.  Ina certain lakethe valueof khas been determined to be the valueyou chose above, which means that 100k% ofthe surfacelight is absorbedeveryfoot ofdepth. For example, ifyou chose0.062, then 6.2%of thelight would be absorbedeveryfoot of depth. Whatis theintensityof lightat a depth of 10 feet if the surfaceintensityis foot candles?  (Correctlyroundyouranswerto one decimalplaceand showtheintermediatestepsin yourwork.)

c.  What is the depth of thephoticzone forthis lake?  (Hint: , so ; solve this equation for x. Correctlyround your answertoonedecimal placeand showtheintermediatestepsin yourwork.)

 

 

 

 

 

Problem2 – CompoundInterest

 

Fordiscrete periods oftime (likeonceperyear, twiceperyear, four timesperyear, twelve times per year, 365 times per year, etc.), the English terms we use to describe these, respectively, are annually, semiannually, quarterly, monthly, daily, etc. The formula for calculating the future amount when interest is compounded at discrete periods of time is:

 

 

A is the amount you will have after t years the money is invested, P is the principal (the initial amount of money invested), r is the decimal equivalent of the annual interest rate (dividethe interest rate by100), and is thenumberof times the interest is compoundedin ONEyear. Forthe compoundingcontinuouslysituation, the formulais:

 

 

A is the amount you willhaveafter t yearsforprincipal, P, invested at r decimal equivalentannual interest rate compounded continuously.

Based on thefirst letter ofyour last name, choosevalues from the table below forPdollars and r percent.

 

Ifyour last name begins

with theletter

Choose an investment amount,

P, dollars between

Choose an interest rate,

r, percentagebetween

A–E $5,000–$5,700 9%-9.99%
F–I $5,800–$6,400 8% -8.99%
J–L $6,500–$7,100 7% -7.99%
M–O $7,200–$7,800 6% -6.99%
P–R $7,800–$8,500 5% -5.99%
S–T $8,600–$9,200 4% -4.99%
U–Z $9,300–$10,000 3% -3.99%

 

Supposeyou invest P dollars at r% annual interest rate. (Correctlyroundyour answers to the nearest wholepenny(two decimal places)and show the intermediate stepsin all these calculations forfull credit.)

a.  Howmuchwillyouhavein8yearsif theinterest is compoundedquarterly?

b.  Howmuchwillyouhavein15yearsif theinterestiscompounded daily?

c.  Howmuchwillyouhavein12yearsif theinterestiscompounded continuously?Use e ≅ 2.718282.

Problem3Newton’sLaw ofCooling

 

AccordingtoSirIsaac Newton’s Law ofCooling,therateat which anobjectcoolsisgivenbythe equation:

 

 

 is the temperature of the object after t hours, is the initial temperature of the object (when t=0),  is the temperature of the surrounding medium, and k is a constant.

 

a.  Supposeadessertatroomtemperature(= 70o F) needstobefrozenbeforeit is served.Thedessertisplacedin afreezerat = 0o F. If thevalue of theconstantis, k = 0.122, what will be the temperature of the dessert after 4 hours? (Use e ≅ 2.718282; correctly round your final answer to two decimal places and show the intermediate steps in your work.)

 

b.  Whatdoyouthink k in thisformula represents?

 

c.  Freezingis 32o F. Howmanyhourswillit takeforthisdesserttofreeze?(Correctly round your   answertotwodecimalplacesandshowtheintermediatestepsin your work.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Problem4–MedicareExpenditures

 

ThefollowingMedicaredata which representstheMedicareexpendituresforyearsafter 2000 intheUnited StatesisfromtheU. S.CensusBureau.

ActualYear Yearsafter2000( x) MedicareExpenditures                    (in BillionsofDollars)
2004 4 311.3
2006 6 403.1
2007 7 431.4
2008 8 465.7
2009 9 502.3

 

A naturallogarithmicregressionfunctionrepresenting this data model is oftheform:

 

 

This data can be closely modeled by the logarithmic regression function:

 

 

a.     Choose a value for x between 15 and 30 (it does not have to be a whole number). Based on this natural logarithmic function, what will be the expenditure for Medicare in the year represented by your chosen value of x? (Correctly round your answer to one decimal place, this is tenths of billions of dollars. Also show the intermediate steps in your work.)

 

b.     Based on this formula, in how many years after 2000 will the Medicare expenditures be $700 billion?  (Correctly round your answer to one decimal place and show the intermediate steps in your work.)

 

c.     Using Excel or another graphing utility and the values from the table above, draw the graph of this function:

On your graph does this data seem to represent a natural logarithmic function? Explain your answer. Is there another function type that we have studied that seems to more closely match the data? Explain your answer.

 

d.     In an English sentence, state the types of transformations of the natural logarithmic function:

 

that will result in the function:

 

Problem5–RichterScale

 

 

 

The Richter scale is a common logarithmic function (base 10) based on a standard energy release ofjoules. To find the Richter scale’s magnitude of an earthquake, M, the energy released by an earthquake, E in joules, is measured against the standard by the formula:

 

 

a.  Basedon thisformula;completethetablebelow. (Correctlyroundyouranswertoone decimalplaceandshowtheintermediatestepsineachof the calculations.) (Hint: for example: rounded to one decimal place.Please see http://www.purplemath.com/modules/exponent.htm for help with exponent rules.

 

 

 

E
 
 
 
 
 

 

 

 

(NOTE: joules was the estimated energy released by the San Fernando, California earthquake in 1971.)

 

b.  According to the U. S. Geological Service (USGS), the second strongest recorded earthquake on Earth since 1900 occurred about 120 kilometers southeast of Anchorage, Alaska on March 27, 1964. The Richter magnitude of that earthquake was registered as 9.2. What would be the energy released in joules of an earthquake of magnitude 9.2? (Correctly round your answer to one decimal place and show the intermediate steps in your work.) (Hint: replace M( x ) by 9.2 and solve the logarithmic equation for x ; then multiply x by to get the value of E for this magnitude.)

 

 

 

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