course MTH 163 6:55 PMSunday, November 29, 2009
.............................................
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): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `qExplain how the rule a(n) = a(n-1) + a(n-2) is related to the enumeration of the rabbit population. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): The fibonnaci sequence has 2 base cases of a(0) = 1 and a(1) = 1. Given these two base cases, the rest of the sequence falls into line by summing the previous two cases (e.g. a(2) = a(1) + a(0) = 1). Thus, for each month n, we can determine the total population of rabbits by summing the populations from month 10 and 11. ------------------------------------------------ Self-critique rating: 3 ********************************************* 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): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* 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: n a(n) / a(n-1) = 1 1/1 = 1 2 2/1 = 2 3 3/2 = 1.5 4 5/3 = 1.667 5 8/5 = 1.6 6 13/8 = 1.625 7 21/13 = 1.615 8 34/21 = 1.619 9 55/34 = 1.618 10 89/55 = 1.618 The graph would oscillate up and down, but the magnitude of each oscillation decreases each time until n = 10 where it becomes approximately the same as n = 9 (due to rounding). confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* 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: For the fibonacci sequence: a(5) = 8 a(10) = 89 Substituting these values of n for x and the resulting values for a(n) for y, we obtain: 8 = a b^(5) 89 = a b^(10) We then divide the equations to obtain our model: 89 = a b^(10) 8 = a b^(5) ------------- 11.125 = 1 b^(10-5) 11.125 = b^5 Then solve for b: 11.125^(1/5) = b^5^(1/5) b = 11.125(1/5) b = 1.619 And substitute for b into one of the original equations to find a: 8 = a (1.619)^(5) 8 = a (11.123) a = 8 / 11.123 a = 0.719 With a and b found, we now have our model: y = 0.719 * 1.619^x confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* 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: For y = .719 * 1.619^x, substituting for x: y = .719 * 1.619^1 = 1.164061 y = .719 * 1.619^2 = 1.884614759 y = .719 * 1.619^3 = 3.051191294821 y = .719 * 1.619^4 = 4.9398787063152 y = .719 * 1.619^5 = 7.9976636255243 y = .719 * 1.619^6 = 12.9482174097239 y = .719 * 1.619^7 = 20.963163986343 y = .719 * 1.619^8 = 33.9393624938891 y = .719 * 1.619^9 = 54.9478278776066 y = .719 * 1.619^10 = 88.960533333845 x model fib dif 1 1.164061 1 -.164 2 1.884614759 2 .115 3 3.051191294821 3 -.51 4 4.9398787063152 5 .06 5 7.9976636255243 8 .002 6 12.9482174097239 13 .052 7 20.963163986343 21 .037 8 33.9393624938891 34 .061 9 54.9478278776066 55 .052 10 88.960533333845 89 .039 avg. dev of diff. magnitudes = -.164 + .115 + -.51 + .06 + .002 + .052 + .037 + .061 + .052 + .039 = -0.0256 confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I used the values for a(1) through a(10) instead of 0 - 9. My apologies. ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `qIf 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: If each mature pair produced 2 pairs of offspring each month, the sequence would be modified like so: a(n) = a(n-1) + 2 a(n-2) month population (in pairs) 0 1 1 1 2 1 + 2 (1) = 3 3 3 + 2(1) = 5 4 5 + 2(3) = 11 5 11 + 2(5) = 21 6 21 + 2(11) = 43 confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* 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: If it took a pair of rabbits 2 months to mature, we would need to modify the difference equation to account for the extra month like so (we would also need another base case to account for the extra month): a(n) = a(n-1) + a(n-3) Instead of summing the last month (n-1) with the month previous to last month (n-2), we have to use the month one month before that one (n-3). confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `qproblems 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To determine dosage D to maintain a required level L of a drug, with a known drug-removal precentage by the kidneys as r, we find that to maintain level L, we must find the proper dosage equal to that level, keeping into account the removal rate of the body, r. Thus, we find that setting level L equal to the unknown dosage D plus the level L quantity multiplied by the amount not removed by the kidneys, or one minus r. So, L = (D + L) (1-r), which can be solved for D to determine the proper dosage: D = (r * L) / (1 - r). confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: 3 `qStarting 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: D = 429 mg L = 1000 mg r = .3 t = 6 hours Exponential function: f(t) = A b^t ... confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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): I was not sure how to fit the data to the function. I see that when t = 0, b^t evaluates to 1, so that f(t) = A * 1 = A. I also see that when t = 6 hours, y must be .7 A. You then set these equal to one another and solved for b, then were able to solve for A when t = 0. ------------------------------------------------ Self-critique rating: 3 ********************************************* 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 = ? L = 800 mg r = .7 D = (r * L) / (1 - r) = (.7 * 800 mg) / (1 - .7) = 560 mg / .3 = 1,866.667 mg confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* 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: Assuming 'maintenance dose' = D, and 'dose of 500mg' = L: L = 500 mg r = .35 D = (r * L) / (1 - r) = (.35 * 500 mg) / (1 - .35) = 175 mg / .65 = 269 mg confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `qIf 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? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: L0 = 0 mg D = 500 mg t = 6 hrs r = .4 L6 = 500mg * (1 - .4) = 300mg confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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): I think the question may have been misworded. I understand how the process the goes for 6 doses rather than 6 hours, though. ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `qWhat sort of function do you think would model the just-before-dose drug concentration? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: exponential confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
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): I was merely making a guess since we were asked for an exponential function in a previous question. ------------------------------------------------ Self-critique rating: 1 ********************************************* Question: `qproblem 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. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: L0 = 0 mg L1 = 750mg * .45 = 338 mg L2 = (338mg + 750mg) * .45 = 490 mg L3 = (490mg + 750mg) * .45 = 558 mg L4 = (558mg + 750mg) * .45 = 589 mg L5 = (589mg + 750mg) * .45 = 603 mg L6 = (603mg + 750mg) * .45 = 609 mg LF = (609mg + 750mg) * .45 = 612 mg confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** 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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating:3 ********************************************* Question: `qWhat is the function that models the minimum concentration during a cycle vs. the number of the cycle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given our minimum concentrations: L0 = 0 mg L1 = 750mg * .45 = 338 mg L2 = (338mg + 750mg) * .45 = 490 mg L3 = (490mg + 750mg) * .45 = 558 mg L4 = (558mg + 750mg) * .45 = 589 mg L5 = (589mg + 750mg) * .45 = 603 mg L6 = (603mg + 750mg) * .45 = 609 mg LF = (609mg + 750mg) * .45 = 612 mg We could create a graph of min. concentration vs number of the cycle: x = num of cycle y = min concentration diff 2nd diff 0 0 --- --- 1 338 338 --- 2 490 152 186 3 558 68 84 4 589 31 37 5 603 14 16 6 609 6 8 7 612 3 3 We know that at some point, if the sequence continues, the dosage will equal the minimum concentration and we won't be able to exceed the minimum concentration level. And I'm not sure where to go from there.... confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** 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 about 614, 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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I'm not sure I could've put that all together without some serious help (or an immense ammount of time).
.............................................
Given Solution: `a** 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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I was a little unsure whether [ 500 mg + a(n - 1) ] should be in brackets for the operation to be performed before taking taking 40% of the figure. ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `qproblem 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 ? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The dosage is 800mg and the proportion removed is 70%. confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `aThe .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. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: `q problem 23 Write the general recurrence relation for initial concentration 0, dosage D and proportion removed = r. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: a(n) = D + a(n - 1) * (1 - r) confidence rating: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: `a** 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). ** Add comments on any surprises or insights you experienced as a result of this assignment. I find the dosage problem to be pretty interesting. It makes me wonder if pharmacists or doctors are aware of the mathematics behind the situation and utilize similar formulas in determining proper dosages for a given patient."