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Math Project 2014 

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MATH133 Individual Project Assignment NAME:_____________________________ (Show your work for these calculations. Review to see how to type mathematics using the keyboard symbols.) Problem 1 – Photic Zone Light entering water in a pond, lake, sea, or ocean will be absorbed or scattered by the particles in the water and its intensity , I, will be attenuated by the depth of the water, x, in feet. Marine life in these ponds, lakes, seas, and oceans depend on microscopic plant life that exists in the photic zone. The photic zone is from the surface of the water down to a depth in that particular body of water where only 1% of the surface light remains unabsorbed or not scattered. The equation that models this light intensity is: I = I 0 e -kx In this exponential function, I 0 is the intensity of the light at the surface of the water, k is a constant based on the absorbing or scattering materials in that body of water and is usually called the coefficient of extinction, e is the natural number e ≅2.718282, and I is the light intensity at x feet below the surface of the water. a. Choose a value of k between 0.025 and 0.095. b. In a certain lake the value of k has been determined to be the value you chose above, which means that 100k% of the surface light is absorbed every foot of depth. For example, if you chose 0.062, then 6.2% of the light would be absorbed every foot of depth. What is the intensity of light at a depth of 10 feet if the surface intensity is I 0 = 1000 foot candles? (Correctly round your answer to one decimal place and show the intermediate steps in your work.) c. What is the depth of the photic zone for this lake? (Hint: I = 0.01 , so 0.01 = e -0.062x ; solve I0 this equation for x. Correctly round your answer to one decimal place and show the intermediate steps in your work.) Page 1 of 5 Problem 2 – Compound Interest For discrete periods of time (like once per year, twice per year, four times per year, 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: r  A = P1 +  n  nt 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 (divide the interest rate by 100), and n is the number of times the interest is compounded in ONE year. For the compounding continuously situation, the formula is: A = Pe rt A is the amount you will have after t years for principal, P, invested at r decimal equivalent annual interest rate compounded continuously. Based on the first letter of your last name, choose values from the table below for P dollars and r percent. If your last name begins with the letter A–E F–I J–L M–O P–R S–T U–Z Choose an investment amount, P, dollars between $5,000–$5,700 $5,800–$6,400 $6,500–$7,100 $7,200–$7,800 $7,800–$8,500 $8,600–$9,200 $9,300–$10,000 Choose an interest rate, r, percentage between 9% - 9.99% 8% - 8.99% 7% - 7.99% 6% - 6.99% 5% - 5.99% 4% - 4.99% 3% - 3.99% Suppose you invest P dollars at r% annual interest rate. (Correctly round your answers to the nearest whole penny (two decimal places) and show the intermediate steps in all these calculations for full credit.) a. How much will you have in 8 years if the interest is compounded quarterly? b. How much will you have in 15 years if the interest is compounded daily? c. How much will you have in 12 years if the interest is compounded continuously? Use e ≅ 2.718282. Page 2 of 5 Problem 3 – Newton’s Law of Cooling According to Sir Isaac Newton’s Law of Cooling, the rate at which an object cools is given by the equation: T = Tm + (T0 − Tm )e − kt T is the temperature of the object after t hours, T0 is the initial temperature of the object (when t=0), Tm is the temperature of the surrounding medium, and k is a constant. a. Suppose a dessert at room temperature ( T0 = 70o F) needs to be frozen before it is served. The dessert is placed in a freezer at Tm = 0o F. If the value of the constant is, 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. What do you think k in this formula represents? c. Freezing is 32o F. How many hours will it take for this dessert to freeze? (Correctly round your answer to two decimal places and show the intermediate steps in your work.) Page 3 of 5 Problem 4 – Medicare Expenditures The following Medicare data which represents the Medicare expenditures for years after 2000 in the United States is from the U. S. Census Bureau. Actual Year Years after 2000 ( x ) 2004 2006 2007 2008 2009 4 6 7 8 9 Medicare Expenditures (in Billions of Dollars) 311.3 403.1 431.4 465.7 502.3 A natural logarithmic regression function representing this data model is of the form: g ( x) = a + b ln( x) This data can be closely modeled by the logarithmic regression function: E( x) = −9.5904 + 229.9582 ln( x) 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: E( x) = −9.5904 + 229.9582 ln( x) 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: f ( x) = ln( x) that will result in the function: E( x) = −9.5904 + 229.9582 ln( x) Page 4 of 5 Problem 5 – Richter Scale The Richter scale is a common logarithmic function (base 10) based on a standard energy release of E0 = 10 4.4 joules. 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:  E M  0.6667 log    E0  a. Based on this formula; complete the table below. (Correctly round your answer to one decimal place and show the intermediate steps in each of the calculations.) (Hint: log( a 10b ) = log( a) + b; for example: log( 5 105.6 ) = log( 5) + 5.6  0.69897 + 5.6  6.29897  6.3, rounded to one decimal place. Please see for help with exponent rules. E x= E E0 0.5  10 6 0.5  101.6 1.0  10 8 1.0  10 3.6 1.5  1010 1.5  10 5.6 2.5 1012 2.5  10 7.6 1.99  1014 1.99  10 9.6 M ( x)  0.6667 log (x) (NOTE: 1.99  1014 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 10 4.4 to get the value of E for this magnitude.) Page 5 of 5 ...
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