#$&* course MTH 163 If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.
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Given Solution: `a** Our assumptions are that rabbits require a month after birth to mature, mature rabbbits produce a pair of newborns every month starting the first month after they reach maturity, and rabbits never die. Every month the number of newborn pairs is equal to the number of mature pairs in the preceding month, which is equal to the total number of pairs from the month before that. Since all the rabbits from the preceding month are still present, the total number in the new month will be equal to the total number from the preceding month plus the number of newborns, which is equal to the total number from the preceding month plus the month before that. If we start with 1 pair of newborns then 1 month later we have a pair of mature rabbits, so after another month we have 2 pairs of rabbits. Our first three numbers are therefore 1, 1 and 2. In the next month we will have the 2 pairs we had in the preceding month plus 1 pair of newborns from the pair we had the month before that for a total of 3 pairs. In the next month we will have the 3 pairs we had in the preceding month plus 2 pairs of newborns from the pair we had the month before that for a total of 5 pairs. In the next month we will have the 5 pairs we had in the preceding month plus 3 pairs of newborns from the pair we had the month before that for a total of 8 pairs. The pattern continues 8 + 5 = 13 13 + 8 = 21 21 + 13 = 34 etc. . ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qExplain how the rule a(n) = a(n-1) + a(n-2) is related to the enumeration of the rabbit population. A(n) would be equal to the total number of rabbits, (n) would be the month. A(n-1) would be total for the previous month and then a(n-2) is equal to the total for the prior 2 months. ------------------------------------------------ Self-critique rating: ********************************************* Question: `qSTUDENT ANSWER: a(n) is total number of rabbits for month(n) a(n-1) is total for previous month a(n-2) is total for 2 months previous which is total mature for 1 month previous INSTRUCTOR COMMENT: Good. In terms of the solution given for the preceding exercise: In the nth we will have the a(n-1) pairs we had in the preceding month plus a(n-2) pairs of newborns from the a(n-2) pairs we had the month before that for a total of a(n) = a(n-1) + a(n-2) pairs. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qWhat are your ratios a(n) / a(n-1) for n = 1, 2, 3, 4, ..., 10, and what does your graph of ratio vs. n look like? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 1, 2, 1.5, 1.66, 1.6, 1.625, 1.615, 1.619, 1.6176 1.6181 1.61797 are the ratios. It would be a jagged graph. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The ratios are 1, 2, 1.5, 1.66, 1.6, 1.625, 1.615, 1.619, 1.6176 1.6181 1.61797 etc. The graph is jagged, up one time, down the next, but jumping less and less each time. If you were to make a horizontal line through two successive graph points, all subsequent points would be 'squeezed' between these lines. COMMON ERROR: The ratios are 1, 2, 1.5, 1.7, 1.6, 1.6, 1.6, ... . You rounded off before you could see the difference in the last 3 results. If you don't see a difference in the ratios you need to use more significant figures--carry your calculation out to enough decimal places that the differences show. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q problem 8 Use the Fibonacci sequence data points for n = 5 and n = 10 to obtain an exponential model in the form y = a b^x. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 8 = A b^5 89 = A b^10 11.125 = b^5, b = (11.125)^1/5 = 1.619. 8 = A * (1.619)^5 A= .711. y = .719 * 1.619^x confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** You would use the points (5, 8) and (10,89) for your y = A b^x model. You would get the equations 8 = A b^5 89 = A b^10 Dividing the second equation by the first you would get 11.125 = b^5, so b = (11.125)^1/5 = 1.619. Substituting into the first equation we get 8 = A * (1.619)^5 A = 8 /(1.619)^5 = .711. So the model is y = .719 * 1.619^x. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qWhat are the values predicted by your model for the first 10 members of the Fibonacci sequence? What is the average error of your approximation? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 0.719, 1.163342, 1.882287356, 3.045540942, 4.927685244, 7.972994725, 12.90030547, 20.87269424, 33.77201928, 54.6431272 The average error is -.017 confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** For x = 0, 1, 2, ..., 9 the formula gives us values 0.719, 1.163342, 1.882287356, 3.045540942, 4.927685244, 7.972994725, 12.90030547, 20.87269424, 33.77201928, 54.6431272. These numbers differ from the Fibonnaci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 by -0.281, 0.163342. -0.117712644, 0.045540942, -0.072314756, -0.027005275, -0.099694535, -.127305757, -0.22798071, and -0.356872798. These are the errors in the function. The average of these errors is easily found by adding the errors and dividing by 10, obtaining an average of about -.016. ** YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I got -0.017 for my average error. I hope this is close enough. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aIf mature rabbits were to produce two baby pairs per month, what then would be the definition of the Fibonacci-type sequence that models the situation where we start with 1 pair of baby rabbits, and what would be population be at the end of each of the first 6 months?{}{}** Remember that the total number of pairs in month n-2 will be the number of mature pairs in month n - 1, which will produce the newborns in month n. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 1 + 2 * 1 = 3 3 + 2 * 1 = 5 5 + 2 * 3 = 11 11 + 2 * 5 = 21 21 + 2 * 11 = 43 43+2*21=85 and so on confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe model would give 2 baby pairs for each mature pair; in month n the number of mature pairs is a(n-2), so there would be 2 a(n-2) baby pairs in month n. The a(n-1) pairs from the previous month would also survive, giving a population of a(n) = a(n-1) + 2 a(n-2). The numbers would be 1, 1, 3, 5, 11, 21, 43, ...... . e.g, 1 + 2 * 1 = 3; 3 + 2 * 1 = 5; 5 + 2 * 3 = 11; 11 + 2 * 5 = 21; 21 + 2 * 11 = 43 etc.. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qHow would the Fibonacci model change if rabbits required two months to mature, with each mature pair still producing 1 pair of baby rabbits per month?ith each mature pair still producing 1 pair of baby rabbits per month? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The number of baby pairs in month n would be the number of mature pairs from the month before, which would be the total number of pairs from 2 months previous to that, so we would be looking back 3 months to the population a(n-3). There would be a(n-3) mature pairs in month n-1 which would give a(n-3) baby pairs in month n. The rabbits from month n-1 would still survive so we would have a(n) = a(n-1) + a(n-3). Note that to get started with this model we would need the populations for the first three months. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): This one I am not sure about. Does it go like the others, but just the first numbers change or the numbers are pushed back? ------------------------------------------------ Self-critique rating:
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Given Solution: `a** If proportion r has been removed to give concentration L, then 1-r of the amount immediately after the dose will remain. The amount immediately after the dose is L + D so we have (L + D) * ( 1 - r) = L. You then solve this equation for L: Expanding the left-hand side we have L(1-r) + D(1-r) = L, or L - r L + D - r D = L. Rearranging the equation to isolate L we get - r L = - ( D - r D), or L = ( D - r D) / r = ( 1-r) / r * D. ** YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I didn’t go further past the first equation confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aStarting with a dose of 429 mg at the instant the concentration of a drug reaches 1000 mg, and knowing that 30% of the drug is removed in 6 hours, what exponential function models the amount of drug vs. time for the 6-hour interval between doses? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When t=6, y is 30% less than A so y =0.7A. .7 A = A b ^ 6 and b = .7 ^(1/6) = .942 y = A * .942 ^ t is the model confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** We need an exponential function y = A * b^t (note that when t = 0 y = A) such that when t = 6, y is 30% less than A, or y = .7 A. Thus .7 A = A b ^ 6 and b = .7 ^(1/6) = .942. Our model is thus y = A * .942 ^ t. Since when t = 0 we have y = 1429 mg, the model is y = 1429 mg * .942^t. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qWhat dosage is required to maintain a minimum level of 800 mg if 70% of the drug is removed between doses? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (D + 800 mg) ( 1 - .7) = 800 mg D = 560 mg / .3 = 1866.6 mg confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The minimum level occurs just before the next dose, and must be 70% of the maximum level which occurs just after the dose. If D is the dose and 800 mg is the level just before the does, then just after the dose the level is 800 mg + D. This level falls to 70%, i.e., to .70 ( 800 mg + D), just before the next dose. This level should be 800 mg. We therefore want ( D + 800 mg) ( 1 - .7) = 800 mg. Subtracting 1 - .7 to get .3, and multiplying this through the grouped expression on the left we get .3 D + 800 mg * .3 = 800 mg so .3 D = 800 mg - .3 * 800 mg = 560 mg and D = 560 mg / .3 = 1867 mg (check my arithmetic). ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qWhat will be the maintenance dose of a drug, given a dose of 500 mg with 35% of the drug removed between doses? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The way this question is stated, with the word 'dose' used for two different quantities, is incorrect. With this miswording you have the choice of lett 500 mg stand for the dose, or for the maintenance level. If 500 mg is read as the dose, then the maintenance level is L = ( 1-r) / r * D = (1 - .35) / .35 * D = .65 / .35 * D = 930 mg, approx.. If you read this as a 500 mg maintenence level then we L = 500 mg and r = .35 so we have L = ( 1-r) / r * D so that D = r / (1-r) * L = .35 / (1-.35) * 500 = .35 / .65 * 500 = 269.23. This is the dose required to maintain a 500 mg level. If you read this as saying that I believe you might have added the .3 * 800 mg to both sides, which would have given you something a lot like the 3467 mg you got. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I am very very confused on this one. ------------------------------------------------ Self-critique rating:
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Given Solution: `a** In the time between doses concentration will fall to .60 of the after-dose concentration. Just before the second dose the level will be .60 * 500 mg = 300 mg. Just after the second dose the level will be 300 mg + 500 mg = 800 mg. Just before the third dose the level will be .60 * 800 mg = 480 mg. Just after the third dose the level will be 480 mg + 500 mg = 980 mg. Just before the fourth dose the level will be .60 * 980 mg = 588 mg. Just after the fourth dose the level will be 588 mg + 500 mg = 1088 mg. Just before the fifth dose the level will be .60 * 1088 mg = 653 mg approx.. Just after the fifth dose the level will be 653 mg + 500 mg = 1153 mg. approx. Just before the sixth dose the level will be .60 * 1153 mg = 692 mg approx.. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `qWhat sort of function do you think would model the just-before-dose drug concentration? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Exponential confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The increases in mg are 180, 108, 65, 39, 24. 108/180 = .6; 65/108=/6; 39/65=.6; 24/39=.6. The ratio sequence for the increases is constant. This implies an exponential function. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: Add comments on any surprises or insights you experienced as a result of this assignment. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: Add comments on any surprises or insights you experienced as a result of this assignment. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!