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H? HF I I I I I I I I J J J  J  J J J J JV J J J J J J J J J J J J J J J J J Ky K K K K K L/ M# M+ -<-ߠ- ---,---#-'-h+-H/-t3-d7-x;-߈?-ߘCM- GM-߀IN-߼ UNIT 14: SOLVING PROBLEMS WITH POWERS and ROOTS ODD QUESTIONS 1-39 1. Match the description on the left with the correct expressions on the right. a. An exponent of 1.57 1) y x b. A base of y 2) 53 c. The x th root of y 3) 5 3 d. Five cubed 4) 51.57 e. 1.57 to the 5th 5) x y f. The fifth root of 3 6) 1.575 3. An astronomer has noted that we can see about 100 quintillion (1020) stars with our telescopes on earth. This astronomer estimates that about 1 trillionth (10-12) of these stars should have planets that are similar to Earth. a. Without using your calculator, multiply the number of stars by the fraction that might have earth-like planets to find the estimated number of stars with earth- like planets. b. Perform the same multiplication with your calculator to check your work in Part a. 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): 5. Calculations involving probability require calculations of powers. Throwing dice is a common example found when playing many games. Since there are six equally possible results from throwing one die, the probability of throwing a six, for exampl, is 1/6. You would expect to obtain a six ounce every 6 throws. When throwing two dice, there are 36 (62) possible results. Just one of these possible results is for both dice to show a six -- 35 of the possible results are not a pair of sixes. Thus the odds against throwing a pair of sixes are 35 to 1. You would expect to obtain a pair of sixes only once every 36 (62) throws. a. Write a formula for computing the number of possible results when throwing n dice: six to the n th power. b. Compute the number of possible results when throwing three dice. What are the odds against throwing all sixes? Hint: Write the ratio of number of ways of not throwing three sixes to the number of ways that you can throw three sixes. c. Complete the following sentence: When throwing 6 dice, you would expect to obtain all sixes once every throws of dice. 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): 7. You are estimating the amount of wood on a log truck using the Doyle log rule. To use the rule you measure the diameter of the small end of a log (in inches) and its length (in feet), then use the formula BF = 1 x (d - 4)2 4 where BF is the approximate board feet of usable lumber in a log, d is the logs small-end diameter in inches, and is the length of the log in feet. Estimate the number of board feet on a log truck carrying 20 logs that are each 27 feet long with a small-end diameter of 16. 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): 9. When raising breeding stock, you may want to calculate the percent homozygosity using the formula % homozygosity = (2m - 1)n x 100% 2m where m is the number of gererations and n is the number of gene pairs Compute the % homozygosity for three gene pairs after five generations. 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): 11. When irrigating farmland, the rate at which the water will infiltrate, or soak into the soil, must be considered. The soil on your farm land currently has an infiltration rate given by the formula I = 3.5 x (T + 1) (-0.64) where I is the rate of infiltration in inches per hour, and T is the time in minutes since the irrigation began. a. Make a table of infiltration rates for various times (for example, every 10 minutes) from 0 minutes to 60 minutes. b. Make a graph with the time since the start of irrigation as one axis, and the infiltration rate as the other axis. Sketch a curve (that passes through your plotted data) that seems to describe the changing rate. You may want to add some more values to your table and plot them, too. c. What is the infiltration rate when you first begin irrigating, at T = 0? d. Experience has shown that when the infiltration rate reaches 0.3 inch per hour, runoff will occur. From the graph, determine about how long after you begin irrigation that runoff will begin. 13. A bank offers a savings account at 6% annual interest rate, compounded annually. Suppose you deposited $100 in this account and left it untouched. After one year, the account would have $100.00 plus the $6.00 interest earned (6% x $100), or $106.00. After the second year, the account would have the $106.00 plus the $6.36 interest (6% x $106), or $112.36. You can continue this process by using the formula Balancen = Deposit x (1 + i )n where n is the number of years the deposit is left in the bank, i is the annual interest rate (in decimal form), Balancen is the initial deposit made to the account after n years, and Deposit is the initial deposit made to the account. What is the balance in the account after 10 years of leaving it to accumulate interest? after 20 years? 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): 15. The L-Mart Corporation is opening stores in a new territory. The Marketing Division has provided an equation for projected revenue for the next five years for the area, as shown below. The equation is based on a computer prediction using statistics from the area and past experience. R = 35y - 17y 2 + 15y 3 - 2y 4 where R is the projected revenue for the area in $1000 and y is the years from the opening of the new territory, from y = 0 to y = 5 a. What does the equation predict the revenue from the area will be at the end of each of the first five years? b. What might happen if you were to try to use this equation to predict the revenue for the7th year? 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): 17. Suppose you are considering the purchase of a new car. One dealership offers to finance a $12,000 car at 12% interest (1% monthly interest) for 48 months (no down payment). Another dealership offers to finance a similar car costing $13,000 at 10.5% interest (0.875% monthly interest) for 60 months (no down payment). a. Use the formula to determine what the monthly payment would be for each of these offers. Be sure to use the monthly values for the interest and the number of payments. 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 b. What is the total of all the payments for each of the offers? 19. Suppose a growing fast-food chain aspires to triple its number of stores each year and spread across the country. It starts with 1 store. At the end of the first year the chain hopes to triple its number of stores, to a total of 3. The next year the chain hopes to triple its number of stores, to a total of 9 stores, and so on. a. Write an equation to calculate the total number of stores at the end of the 2nd year and at the end of the 3rd year. Write the formula for the total number of stores at the end of the n th year. b. How many stores would the chain have after the 3rd year? after the 6th year? c. Suppose the goal of the company president is to have 50,000 stores across the country. Describe a method you could use to estimate how many years it would take to reach this goal. 21. An annual inflation rate of 6% means that the price of goods (on the average) has increased by 6% by the end of a year. Suppose our country was able to hold its annual inflation at 6%. At the end of the first year the price of goods would be 6% higher, or 1.06 times the initial price. At the end of the second year it would be 12.36% higher than at first, or 1.06 x 1.06 times the initial price. a. Write an equation for the cost of goods after n years of 6% annual inflation: the initial cost times 1.06 raised to the n th power. b. Suppose a loaf of bread initially costs $0.89. How much would you expect it to cost after 10 years at this same inflation rate? after 20 years? 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): 23. Researchers have found that the height and weight of the human body is a valuable factor when studying race, geography, physiological function , and disease. A typical measure is the ratio of the height to the cube root of the weight. Determine your height in meters and your weight in kilograms, and compute the value of this ratio for you. 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): 25. Laboratory technicians often prepare cultures to help identify various bacteria that are present in samples from patients. A culture permits the bacteria to reproduce and create large, more easily identifiable populations. A certain bacterium is able to reproduce every 20 minutes in a culture. Thus in twenty minutes, one bacterium becomes two. After the second twenty minutes, the two bacteria become four. After the third twenty minutes, the four become eight, and so on. a. Write a formula that describes the number of bacteria that will be present after the n th period of 20 minutes: two to the n th power. b. If nothing happens to change the reproduction rate or kill any of the bacteria, how many bacteria will be present after 2 hours? after 5 hours? c. Describe a method that could be used to determine how long it would take for the population of this type bacteria to reach one million in the culture. 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): 27. When caring for patients with radioactive implants, the health care personnel should remain with the patient no longer than necessary. One of the factors controlling the amount of radiation the health care worker receives is the distance from the patient. The radiation level follows the inverse-square law, which says that the intensity is inversely proportional to the square of the distance from the source. If the radiation has an intensity of 100 radiation units at a distance of 1 m, then at a distance of z meters, the intensity I is given by the formula I = 100 radiation units x z -2 Make a table of values for I for distances ranging from 1 meter to 10 meters, in steps of 1 meter. Draw a graph of these intensities. At about what distance does the radiation intensity reach a level of 50 radiation units? a level of 10 radiation units? a level of 1 radiation unit? 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): 29. It is important that safety engineers monitor the sound levels in industrial environments. Measuring devices indicate the noise level in decibels, or dB.  The intensity of a sound (relative to a reference level) is 10 raised to the power of one-tenth the decibel level. So if the sound level in dB is S , then the intensity of I is I = I r x 10(S / 10) where I r is a reference sound level - the threshold of human hearing. Use what youve learned about dividing powers to answer the questions that follow. a. Ordinary speech can produce a measurement of about 50 dB, while a power lawn mower can produce noise as loud as 90 dB. How much louder is the lawn mower than normal speech? (Hint: Compute the ratio of the intensity of the sounds.) b. The Occupational Safety and Health Administration (OSHA) describes permissible sound-level exposures for noisy workplaces. For example, OSHA may recommend no more than 8 hours exposure to an average level or 90 dB, but restrict employees to no more than 3 hours of 97 dB level noise. How much more intense is the sound at 97 dB than the sound at 90 dB? How does this relate to the recommended maximum exposure times? 31. Electrical resistors are coded with colored bands to indicate the value of resistance in ohms. Each color represents a number: Color Number Black 0 Brown 1 Red 2 Orange 3 Yellow 4 Green 5 Blue 6 Violet 7 Gray 8 White 9 The resistance value is determined by obtaining the first and second digits from the first and second color bands, and multiplying by 10 raised to the n th power, where n is the number represented by the third color band. For example, a resistor banded red-brown-orange would have a resistance of 21 (from the red and brown bands) x 103 (from the orange band) ohms, or 21,000 ohms. Determine the resistance values for the following color band combinations. a. Green-brown-orange b. Red-red-blue c. Black-gray-black d. Green-black-black e. Yellow-green-brown 33. Race car drivers and designers must concern themselves with air resistance. Some of the power generated by the engine must be used to overcome this resistance. A formula can be used to estimate the horsepower required to overcome air resistance for a car with a given frontal area, as follows: Power = (s3 x A ) 150,000 where Power is in horsepower, s is the cars speed in miles per hour, and A is the cars frontal area in square feet. Evaluate the power required to overcome air resistance for a car that has a frontal area of 80 square feet, traveling at a speed of 65 mph and at 130 mph. Does twice the speed require twice the horsepower to overcome air resistance? 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): 35. You are using a Type J thermocouple to measure the temperature of a pressurized steam-heated press. You record a voltage of 0.025 volt. Use the calibration formula shown below to compute the temperature indicated by the thermocouple. T = -0.48868252 + 19,873.14503 V - 218,614.5353 V 2 + 11,569,199.78 V 3 - 264,917,531.4 V 4 + 2,018,441.314 V 5 where T is the temperature of the thermocouple in oC, and V is the thermocouple voltage. 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): 37. A photomultiplier tube is used to measure extremely low levels of light. When a single light ray (called a gamma ray) strikes the end of the tube, one or more electrons can be emitted. These electrons are directed to a charged surface where they each cause perhaps 3 to 30 more electrons to be emitted. These then travel to a second surface where, again, each can cause perhaps 3 to 30 more electrons to be emitted. This process can continue for as many as 10 surfaces, so that at the final surface a sizeable avalanche of electrons occurs and can be measured as a brief current pulse. a. Suppose that such a device detects a single ray of light and the first surface is struck by just one electron. This first surface responds by producing 20 electrons. These 20 electrons are directed to the second surface. The second surface responds similarly, producing 20 electrons for each one that strikes it, and so on, up to the tenth surface. Write a mathematical expression for the total number of electrons produced by the ten surfaces and evaluate it. b. If each electron carries a charge of 1.6 x 10 -19 coulombs (pronounced kool- omes), what is the total charge that comes off of the tenth surface? 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): 39. Computers store numbers in binary, that is, with electrical components that are either on or off. By using several of these on or off components, numbers are stored. Each component stores a bit - it has a value of 1 if it is on, or 0 if it is off. Recall that in the decimal system, each digit represents a power of ten - units (100), ten (101), hundreds (102), thousands (103), and so on. In the binary system, each bit (or digit) represents a power of two - 20, 21, 22, 23, and so on, as shown below. BINARY NUMBER PLACE VALUES: ___ 27 26 25 24 23 22 21 20 a. What is the value represented by each place in a binary number, as shown above? b. If the places in such a binary number were all filled with the bits on 1 1 1 1 1 1 1 1 what would be the value of the number stored? (Hint: This is the largest number that can be stored as eight bits in a computer, known as a byte.) 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-h>"9 *6*>"9DSET(H-|9>96*=9DSETR -߰---ވ   -