course Mth 163 Appologies for this being late, I was positive I had run the Query and submitted it before Thanksgiving then realized I had not. CeqX{N
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13:07:30 Give the rabbit populations for the first 12 months. Explain how each new population is obtained, and what your method for obtaining the new population has to do with the assumed nature of rabbit reproduction.
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RESPONSE --> for the first 12 months the rabbit populations are 1,1,2,3,5,8,13,21,34,55,89,144,233. The rules used for this problem are that rabbits need one month to mature and once mature a rabbit pair produce a newborn, and the rabbits never die. With this you can find the populations for months to come, also there are some patterns that help out, the number of mature rabbits at the end of the month is equal to the total number of rabbits at the end of the previous month. Then number of baby rabbits at the end of the month is equal to the number of mature rabbits at the end of the previous month. So the number of rabbits at the end of the month is the number of mature rabbits plus the number of baby rabbits.
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13:09:53 ** 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. . **
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13:12:43 Explain how the rule a(n) = a(n-1) + a(n-2) is related to the enumeration of the rabbit population.
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RESPONSE --> a(n) = a(n-1)+a(n-2) The number a(n) of rabbits at the end of month (n) is equal to the number of rabbits from the previous month a(n-1) plus the number of rabbits from two months ago a(n-2)
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13:14:06 STUDENT 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.
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13:17:21 What 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?
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RESPONSE --> ratios 1,2,1.5,1.6666,1.6,1.625,1.615385,1.619048,1.617647,1.618182 The graph is zig-zag-ish, it goes up and down up and down but it does this less and less each time.
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13:17:29 ** 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. **
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13:22:49 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.
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RESPONSE --> (5,8) -> 8=Ab^5 (10,89) -> 89 = Ab^10 Divide these equations to get 11.125=b^5 raise both sides to 1/5 power to eliminate the power of five on b so 11.125^1/5 = b, b = 1.619 Put this value of b into the equation 8=Ab^5, 8=a(1.619)^5, 8=a(11.123), divide both sides by 11.123, a = .719 Model : y=.719(1.619)^x
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13:25:35 ** 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. **
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13:39:04 What are the values predicted by your model for the first 10 members of the Fibonacci sequence? What is the average error of your approximation?
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RESPONSE --> When x = 0 through 9, the model gives y values .719, 1.164061, 1.884615, 3.051191, 4.939879, 7.997664, 12.948217, 20.963164, 33.939362, 54.947828, 88.960533 Differences from Fibonacci sequence respectively : -.281, .164061, -.115385, .051191, -.060121, -.002336, -.051783, -.036836, -.060638, -.052172, -.039467 My average error was -.044044 found by adding differences together and dividing by 11 (since i used x values 0 - 11)
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13:39:29 ** 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. **
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13:42:38 If 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?
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RESPONSE --> The equation would be changed to a(n-1)+2a(n-2)=a(n) to accomidate for the 2 baby rabbit pairs instead of one. So for the months 0-6 the populations would be 1,1,3,5,11,21,43 respectively
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13:43:06 ** 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.
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13:43:17 The 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.. **
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13:57:07 How 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?
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RESPONSE --> Instead of a(n-2) in the sequence it would have to be a(n-3) since you would have to look three months back in the population instead of two to add to a(n-1). Model would be a(n) = a(n-1)+a(n-3)
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13:59:17 ** 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. **
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14:02:09 problems 12-16 How did you describe how to reason out the dosage D required for a desired level L of a drug if proportion r of the drug present immediately after a dose has been removed at the time of the next dose.
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RESPONSE --> L stands for the desired maintence level, D for dose and r for portion lost between doses. When a dose has been taken, the dosage in the body will be L+D. When time has passed, this does deteriorates to leave 1-r behind so (1-r)(L+D)=L, you do not want to build on an increased amount of chemicals in the bloodstream.
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14:03:07 ** 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. **
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RESPONSE --> I did not solve the equation (L+D)(1-r) = L for L, but the process is understood.
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14:07:11 Starting 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?
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RESPONSE --> Start: 429 mg when concentration is 1000mg 30% removed in 6 hrs. exponential function : y=Ab^t t=6 y=.7A b=.7^(1/6) = .9423 y=A(.9423)^t y=(1000+429mg)(.9423)^t = 1429mg(.9423)^t
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14:07:39 ** 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. **
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14:11:02 What dosage is required to maintain a minimum level of 800 mg if 70% of the drug is removed between doses?
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RESPONSE --> (L+D)(1-r)=L 800mg=(800mg+d)(1-.7) 800mg=(800mg+D)(.3) 800mg=240mg+.3D560mg=.3D D= 1866.67mg
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14:12:08 ** 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). **
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14:14:25 What will be the maintenance dose of a drug, given a dose of 500 mg with 35% of the drug removed between doses?
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RESPONSE --> L=(1-r)/r*D L=(1-.35)/.35*500 L=(.65/.35)*500 L=928.571mg
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14:15:15 ** 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. **
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14:18:19 If a patient starts with no drug in her body and takes a 500 mg dose every six hours, losing 40% of the after-dose amount during that time, then how much drug will remain six hours after the initial dose?
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RESPONSE --> Start:0 500 mg dose every 6 hrs losing 40% (1-.4=.6) How much remains six hours after initial dose? .6*500 = 300 mg before 2nd dose 300+500=800mg after 2nd dose. .6*800 = 480 mg before 3rd dose 480+500 = 980 mg after 3rd dose. .6*980 = 588 mg before 4th dose 588+500 = 1088 mg after 4th dose .6*1088=652.8 mg before 5th dose 652.8+500 = 1152.8 mg after 5tth dose .6*1152.8=691.68 jut before 6th dose.
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14:18:49 ** 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.. **
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14:19:55 What sort of function do you think would model the just-before-dose drug concentration?
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RESPONSE --> Differences between values are 180, 108, 64.8, 38.88, second differences are all 1.6666.... constantly, so an exponential function would work as a satisfactory model.
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14:20:24 ** 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. **
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RESPONSE --> it is a ratio sequence instead of the second difference as I said.
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14:32:26 problem 19 Determine the minimum concentration vs. dosage cycle of a drug for six dosage cycles, starting with the first dose, if the dosage is 750 mg and if 55% of the maximum concentration of the drug is removed during each cycle.
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RESPONSE --> 6 dosage cycles 750 mg dose 55% removed each cycle (1-.55=.45) Minimum concentration for first 6 doses respectively 0, 337.5, 489.375, 557.719, 588.473, 602.313
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14:33:02 ** After 55% of the first 750 mg dosage is removed there will be about 338 mg left; after the next 750 mg dose there will be 1088 mg, which after removal of 55% will leave about 490 mg. etc. This gives us minimum concentrations of about 338 mg, 490 mg, 558 mg, 588 mg, 602 mg, 609 mg, and 611 mg. **
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14:53:32 What is the function that models the minimum concentration during a cycle vs. the number of the cycle?
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RESPONSE --> minimum concentration= 615.076(1-.449^n)
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14:54:11 ** As before you can find that the ratios of the sequence of differences are constant, so the function exponentially approaches the limiting concentration. The limiting after-dose concentration L is that for which the amount removed during a cycle is equal to the dosage. At that level the amount removed will be .55 L, which will equal the 750 mg dose. So we have .55 L = 750 mg, which gives us L = 750 mg / .55 =1363.6 mg approx.. The minimum concentration will be 1363.6 mg - 750 mg = 613.6 mg, approx.. Our concentrations of 0, 338, 490, 558, 588, 602, 609, 611 mg lie 276, 124, 56, 25, 11, 5 and 2 mg below the limiting minimum concentration. We find the exponential function that models these differences. The differences are modeled by an exponential function y = A b^n, with y = 276 when n = 1 and y = 124 when n = 2. Solving the equations 276 = A * b^1 124 = A * b^2 for A and b we obtain A = 613.6 and b = .45. The function for the differences is therefore difference = 613.6 * .45^n. The concentrations lie below the limiting minimum concentration of 613.6 by this amount, so our exponential function is concentration = 613.6 - 613.6 * .45^n, or if we factor out the 613.6 concentration = 613.6 ( 1 - .45^n), where the concentration is in mg and n stands for the number of the last dose taken. **
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15:02:37 problem 21 If the dosage is 500 mg and the proportion removed during a dosage cycle is .6, then in terms of a(n-1), write an expression for the drug concentration just after dose number n-1
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RESPONSE --> dose = 500mg .6 removed (1-.6=.4) a(n) = .4*a(n-1)+500
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15:02:52 ** If we have a(n-1) just after dose n - 1 then just before dose n we have .40 * a(n-1). So just after dose n we will have a(n) = .40 * a(n-1) + 500 mg. This recurrence relation together with the initial condition a(0) = 0 defines the situation. }If a(0) = 0 then we have a(1) = .40 * a(0) + 500 mg = 500 mg a(2) = .40 * a(1) + 500 mg = .40 * 500 mg + 500 mg = 700 mg a(3) = .40 * a(2) + 500 mg = .40 * 700 mg + 500 mg = 780 mg etc. **
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15:07:49 problem 22 What is the dosage and what proportion of a drug is removed if the recurrence relation for the concentration a(n) just before the nth dose is a(n) = .3(a(n-1) + 800), a(1) = 0 ?
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RESPONSE --> There is an 800 unit dose and 70% of the drug is removed between doses (.3 or 30% remains so 1-.3=.7 or 70%)
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15:08:00 The .3 indicates that .3 of the drug remains between doses so that .7 or 70% is removed. The 800 indicates that a does of 800 units is added to the existing concentration with each new does.
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15:11:07 problem 23 Write the general recurrence relation for initial concentration 0, dosage D and proportion removed = r.
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RESPONSE --> a(n) = D +(1-r)*a(n-1)
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15:11:25 ** If a(n) stands for the amount just after dose number n then (amt just after dose n) = dose + (amt after proportion r is removed from dose n - 1) we have a(n) = D + (1 - r) * a(n-1). **
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15:11:26 Add comments on any surprises or insights you experienced as a result of this assignment.
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