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Make a table of f- stops for various distances from 5 feet to 100 feet. b. Make a graph of the data in your table, showing the f-stops for various distances. Draw a curve to represent this formula. c. Use the graph, or your table of values: what is the maximum distance you could safely use this flash with an f-stop of 2? 4. Pendulum clocks can maintain accurate time because they have a finely adjustable period of swing. The period of swing of a pendulum is given by the formula T = 2 x L g where T is the period (see note below) of the pendulum in seconds, L is the length of the pendulum in centimeters, and g is the acceleration due to gravity, 980 cm/sec2. Note: The period is defined as the time required for one complete swing, back and forth. a. What would be the period of a pendulum that was 25.00 cm long? (Use your calculators value for with this formula.) b. Suppose it is desired to have the pendulum swinging with a period of 1.00 second. If the length was shortened to 24.90 cm, would the period be closer to or farther from the desired value when compared to the original setting of 25.00 cm? 6. Without insecticides, the farmer would have to watch some forms of insect populations triple with each life cycle of the insect. Suppose a certain area has 8 grasshoppers per acre. After one life cycle these could produce 8 x 3 grasshoppers. After two life cycles, there would be (8 x 3), or 8 x 32 grasshoppers, and so on. a. Write a formula that calculates the number of grasshoppers after n generations: eight times three to the n th power. b. Compute how many grasshoppers would be produced after 10 generations (assuming nothing inhibits their reproduction). What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 8. When estimating the forest-land value, a formula such as shown below is often used. V = N [(1 + I )t - 1] where V is the land expectation value in dollars per acre ($/A), N is the net income received at rotation age ($), I is the interest rate expressed as a decimal, and t is the length of rotation in years. Determine the land expectation value for a pine forest based on a rotation of 60 years, an interest rate of 5%, and a net income of $210 at rotation age. What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 10. Sometimes soil becomes accidentally contaminated by an unwanted chemical. However, microorganisms are often able to decompose some contaminants and reduce the contamination level to about 60% of the existing level every 20 days. Suppose a soil test reveals a contamination of 64,000,000 units per cubic meter of soil. a. Write the formula for determining the contamination level after n 20-day periods: the initial contamination times the quantity 0.06 raised to the n th power. b. What is the level of contamination after 80 days? after 200 days? after 360 days? c. Suppose a safe level is 40,000 units per cubic meter of soil. Describe a method you could use to estimate how long it would take for the soil to reach this safe level. What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 12. When breeding swine, animal identification by ear-notching is the recommended practice. One or two notches (or no notches) in each zone of the right ear are used to yield a sum used to indicate the litter from which the pig came. The left ear is used similarly to identify each pig in the litter. The zones are shown in the drawing below (picture a) with the value for each notch. Each zone can contain 0,1, or 2 notches. For example, the zone labeled 3 can contribute either 0, 3 (when notched once), or 6 (when notched twice) to the total. The zone labeled 9 can contribute either 0, 9 (when notched once), or 18 (when notched twice) to the total. The total litter identification for a pig is then the sum of the values for each zone. ID = Zone0 + Zone1 + Zone2 + Zone3 + Zone4 a. Each zones label in the illustration is really a power of 3, since each zone can have three possible values (0,1 or 2). Identify the exponent of 3 used for each zone. b. Suppose a pig was notched in the right ear (picture b). Compute theID value for this pig. c. Suppose that you were able to place up to three notches in each zone. Then, each zone could have 0,1,2, or 3 notches. What would be the value of each zones notch with this arrangement? 14. The manufacturer of a new line of computers expects revenues to be $409,600 this month. Based on past history, the projected revenue R for the next 12 months is given by the formula R = $409,600 x 1.5625(t/4) for t ranging from 0 to 12 months Determine the expected revenue in months 2, 6 and 12. What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 16. You agree to sell your car to someone for $2600, and allow them to finance it by paying 1% per month interest on the unpaid balance. You would like to amortize the loan in 24 months (finish receiving payment with 24 equal monthly payments). Use the formula below to determine the monthly payment you should ask for. R = P x i (1- [1+ i ] -n) where R is the monthly payment, P is the loan amount ($2600), n is the number of payments (periods) (24 payments), and i is the periodic interest rate (0.01 per month).  What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 18. When mailing out advertisements from a mailing list, you must consider the expected response rate. You may expect a response rate of 3%, for example. The limits for a 95% confidence interval associated with the response rate can be calculated from the formula below. where L1 is the upper limit of the response (95% of the time), L2 is the lower limit of the response (95% of the time), p is the response rate (a decimal fraction), and n is the number of mailouts. What are the upper and lower limits of this 95% confidence interval if you mail out 2500 flyers and expect a response rate of 3% (that is, 0.03)? What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 20. You may have heard of the seemingly small salary offer of starting one out at $.01 per week, and doubling the weekly salary each week. This means that for the first week you would earn $.01. On the second week you would earn $.01 x 2, or $.02. On the third week your salary would increase to $.01 x 2 x 2, or $.04. On the fourth week, your salary would become $.01 x 2 x 2 x 2, or $.08. a. Write the second, third, and fourth week salaries as an expression involving a power of 2. b. Write a formula for the salary for any week, as a power of 2. c. What would be your weekly salary for the 22nd week of your first year of working with this salary arrangement? Suppose your employer decides at the end of the first half of the year (week 26) that your salary must remain at the present level. Would this be a good offer? What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 22. Your company specializes in 12-month series mailings, where the consumer has the option to cancel the series at any time. The Marketing Department has provided you with a formula to predict the cancellation rate of the consumers, based on prior experience. As consumers cancel their subscriptions to the series, the total numbers of mailings will drop, as described by the formula M = (150,000 x 1.4-t ) + 20,000 where M is the number of mailouts each month and t is the month, numbered 0 through 11. a. Determine the initial number of mailouts at the start of the series for month 0. b. Determine the number of mailouts after the first month , t = 1. What fraction of the initial mailouts (Part a) does the Marketing Department think will cancel the series after receiving the first mailing? (Express your answer as a percent.) c. Determine the number of mailouts still going out after the sixth month, t = 6. What fraction of the initial mailouts (Part a) does the Marketing Department think will cancel the series after receiving the sixth mailing? (Express your answer as a percent.) d. How many mailouts are still going out at the last mailing, when t = 11? e. Suppose you know that the cost of production becomes too expensive if the number of mailouts drops below 25,000. Describe a method to use the formula to predict when it would be time to begin mailing another series. 24. Fruit flies are commonly used to study genetics. They reproduce very quickly - about once every two weeks they will produce hundreds of new flies with each generation. Suppose that in two weeks each of these females produces 200 more flies, half of which are female, and so on. If this continues, the number of flies born during the n th generation (after n periods of two weeks) could be estimated by the formula Number of flies born = 2 x 100n Compute the estimated number of flies that could be born after the 4th generation if the process continues uninhibited. What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 26. Patients with possible thyroid malfunction may undergo testing with radioactive iodine (131I). This form of the element iodine has a half-life of 8.07 days, meaning that every 8.07 days the radioactivity has reduced by one half. Thus, if an injection of 131I has an initial activity of 5000 decays per second, then the activity at a later time t can be found by the formula Activity = 5000 decays per sec x (1/2)(t / 8.07) where t is the time passed from the count of initial activity, measured in days. a. Use the formula to determine what the activity of the 131I is after 8.07 days. b. What is the activity after 14 days? c. Describe a method that you could use to find out how long it would take for the activity to reach a level that was about 5% of the original activity. 28. When working in disease-control centers, the danger of contagious diseases must always be remembered. Suppose that a disease is transmitted from one individual to two others during a period of one week. Each of these two transmits the disease to two more individuals during the next week, yielding a total of four newly infected persons. Suppose such a process continues unabated. How many weeks would it take for a city with a population of 500,000 to all have been infected with the disease? (Hint: Whe7n does the total of the infected persons exceed 500,000?) What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 30. The parents of a newborn child wish to establish a fund for the childs college education. Working with an account that pays 6% interest compounded annually, they would like the account to hold $20,000 after 17 years. The annual payment to the account can be computed using the formula R = S x i (1 + i )n - 1 where R is the annual payment to the account, S is the final value of the account after n payments ($20,000),  n is the number of payments (periods) (17 payments), and i is the interest rate (0.06). What annual payment R should the parents make to the account? What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 32. Some infrared temperature detectors use the fact that energy is radiated from hot black objects, according to the formula R = (5.670 x 10-12) x (273oC + T )4 where R is the total energy radiated in watts per cm2 from a perfect radiator, and T is the temperature in degrees Celsius. a. What would be the total energy radiated from such an object that had a temperature of 300oC? b. What would be the total energy radiated from such an object that had a temperature of 37oC (the same as 98.6oF)? What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 34. The rate of flow of water through a fire hose can be computed with the formula Flow rate = 30 x d 2 x P where Flow rate is in gallons per minute, d is the nozzle diameter in inches, and P is the nozzle pressure in pounds per square inch (psi). What is the flow rate that could be expected from a hose with a 2.5 nozzle diameter and a nozzle pressure of 80 psi? What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 36. Torque strength of a shaft is proportional to the cube of the shaft diameters. Thus Strength of shaft 1 = (Diameter of shaft 1) Strength of shaft 2 (Diameter of shaft 2)3 How much stronger is a 1-inch diameter shaft than a shaft that is 1/2 inch in diameter? (Hint: What is the ratio of the strength of the 1-inch shaft to the 1/2- inch shaft?) What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): EXTRA CHALLENGE: Suppose you presently had a 9/16 shaft and needed to replace it with a shaft that was about twice as strong. What shaft diameter would be needed? 38. Astronomers rank the brightness of stars by means of ratios. A difference in brightness of five magnitudes corresponds to a ratio of brightness of 100 to 1. Thus, a difference of magnitude of 1is equivalent to a ratio of brightness of 100 raised to the one-fifth power. A difference of magnitude of 2 is equivalent to a ratio of 100 raised to the two-fwifths power, and so on. a. Write the expressions for the value of the brightness ratios that corresponds to differences in magnitude of 1,2, 3,4 and 5. Compute the values for these ratios. b. Compare the value of 100 raised to the one-fifth power to the value of the fifth root of 100. Can you make any conclusions? Test your conclusion with some other powers and roots. What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): 40. When computing statistics, one must often evaluate the expression n !, known as n factorial. (You may have such a key on your calculator.) Factorial expressions are simply a decreasing series of numxbers multiplied together. For example, 4! = 4 x 3 x 2 x 1 = 24. The value of 5! is 120 (5! = 5 x 4 x 3 x 2 x 1). However, manually evaluating factorials of even slightly larger numbers (try 15!) becomes very tedious and results in very large numbers. So for large values of n , Stirlings formula is often used, as shown below. n ! (2 n x (n /e )n where e is a constant that has a value of about 2.718. Use Stirlings formula to obtain an approximate value for 30!. If your calculator has an x ! key, enter 30 and press x ! (or 2nd x !) and compare the result with Stirlings formula. What are you looking for? What do you need to answer the question? Write out the methods that you will use to solve the problem and show your work in calculating the answer(s). Your answer(s) are (make sure the units used are correct): DSET|(H->"9 *6*>"9DSET(H-9>96*=9DSETR ----   -Unit 14 Even Questions