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MATH133 Unit 5: Exponential and Logarithmic Functions Individual Project Assignment: Version 2A Show all of your work details for these calculations. Please review this Web site to see how to type mathematics using the keyboard symbols. IMPORTANT: See Question 1 in Problem 2 below for special IP instructions. This is mandatory. 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 the following: 𝐼 = 𝐼0 𝑒 βˆ’π‘˜π‘₯ In this exponential function, I0 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 β‰… 2.718282, and I is the light intensity at x feet below the surface of the water. 1. Choose a value of k between 0.025 and 0.095. 2. In a lake, the value of k has been determined to be the value that you chose above, which means that 100k% of the surface light is absorbed for every foot of depth. For example, if you chose 0.062, then 6.2% of the light would be absorbed for every foot of depth. What is the intensity of light at a depth of 10 feet if the surface intensity is I0 = 1,000 foot candles? (Correctly round your answer to one decimal place, and show the intermediate steps in your work.) Page 1 of 6 𝐼 3. What is the depth of the photic zone for this lake? (Hint: 𝐼 = 0.01, so 0.01 = 𝑒 βˆ’0.062π‘₯ .) 0 Solve this equation for x. Correctly round your answer to one decimal place and show the intermediate steps in your work. Problem 2: Compound Interest For discrete periods of time (once per year, twice per year, four times per year, 12 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 π‘Ÿ 𝑛𝑑 interest is compounded at discrete periods of time is 𝐴 = 𝑃 οΏ½1 + 𝑛� , where A is the amount you will have t years after 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 1 year. For the compounding continuously situation, the formula is 𝐴 = 𝑃𝑒 (π‘Ÿ)(𝑑) , where 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 Choose an investment amount, Choose an interest rate, r, the letter P, between between 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% Page 2 of 6 Suppose that 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 of these calculations for full credit.) 1. Important: By Wednesday night at midnight, submit a Word document containing only your name and your chosen values from the table above for P and r. Submit this in the Unit 5 IP submissions area. This submitted Word document will be used to determine the Last Day of Attendance for government reporting purposes. 2. How much will you have in 8 years if the interest is compounded quarterly? 3. How much will you have in 15 years if the interest is compounded daily? 4. How much will you have in 12 years if the interest is compounded continuously? Use 𝑒 β‰… 2.718282. 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 = π‘‡π‘š + (𝑇0 βˆ’ π‘‡π‘š )𝑒 βˆ’π‘˜π‘‘ , where 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. 1. Suppose that a dessert at room temperature (T0 = 70Β°F) needs to be frozen before it is served. The dessert is placed in a freezer at Tm = 0Β°F. If the value of the constant is k = 0.122, what will the temperature of the dessert be after 4 hours? (Use 𝑒 β‰… 2.718282, correctly round your answer to two decimal places, and show the intermediate steps in your work.) 2. What do you think k in this formula represents? 3. Freezing is 32Β°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.) Problem 4: Medicare Expenditures The following health care data represent health care expenditures for years after 2000 in the United States (U.S. Census Bureau, 2012): Page 3 of 6 Actual Year Years After 2000 (x) Medicare Expenditures (in billions of dollars) 2004 4 311.3 2006 6 403.1 2007 7 431.4 2008 8 465.7 2009 9 502.3 A natural logarithmic regression function model of the form, 𝑓(π‘₯) = π‘Ž + 𝑏 ln(π‘₯), representing these data can be found. These data can be closely modeled by the following logarithmic regression function: 𝐸(π‘₯) = βˆ’9.5904 + 229.9582 ln(π‘₯) 1. 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 health care in the year represented by your chosen value of x? (Correctly round your answer to one decimal place, which is tenths of billions of dollars, and show the intermediate steps in your work. 2. Based on this formula, in how many years after 2000 will the health care expenditures be $700 billion? (Correctly round your answer to one decimal place, and show the intermediate steps in your work.) 3. Using Excel or another graphing utility and the values from the table above, draw the graph of this function, 𝐸(π‘₯) = βˆ’9.5904 + 229.9582 ln(π‘₯). 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. 4. In an English sentence, state the types of transformations of the natural logarithmic function, 𝑓(π‘₯) = ln(π‘₯), that will result in the following function: Problem 5: Richter Scale 𝐸(π‘₯) = βˆ’9.5904 + 229.9582 ln(π‘₯) Page 4 of 6 The Richter scale is a common logarithmic function (base 10) based on a standard energy release of 𝐸0 = 104.8 joules. The energy released by an earthquake, E in joules, is then measured against 𝐸 the standard by the formula, 𝑀 β‰… 0.6667 log �𝐸 οΏ½, to get the Richter scale magnitude of the 0 earthquake, M. 1. Based on this formula, complete the following table. Correctly round your answer to one decimal place, and show the intermediate steps in each of the calculations. (Hint: log(π‘Ž Γ— 10𝑏 ) = log(π‘Ž) + 𝑏; example log(5.0 Γ— 105.2 ) = log(5) + 5.2 β‰… 0.69897 + 5.2 β‰… 5.89897 β‰… 5.9 rounded to one decimal place.) Please see this Web site to for help with exponent rules. E 𝒙= 𝑬 π‘¬πŸŽ 0.5 x 106 0.5 x 101.2 1.0 x 108 1.0 x 103.2 1.5 x 1010 1.5 x 105.2 2.5 x 1012 2.5 x 107.2 1.6 x 1017 1.6 x 1012.2 𝑴(𝒙) β‰… 𝟎. πŸ”πŸ”πŸ”πŸ• π₯𝐨𝐠(𝒙) Note: 1.6 x 1017 joules was the estimated energy released by the San Francisco, California earthquake on April 18, 1906 (Pidwirny, 2010). 2. 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 (Historic Earthquakes, 2014). The Richter magnitude of that earthquake was registered at 9.2. What would be 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 104.8 to get the value of E for this magnitude.) 3. Which intellipath Learning Nodes helped you with this assignment? Page 5 of 6 References Exponents: Basic rules. (n.d.). Retrieved from the Purple Math Web site: Formatting math as text. (n.d.). Retrieved from the Purple Math Web site: Historic earthquakes. (2014). Retrieved from the USGS Web site: Pidwirny, M. (2010). Earthquake. Retrieved from the Encyclopedia of Earth Web site: U.S. Census Bureau. (2012). Health and nutrition. Retrieved from Page 6 of 6
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