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the hormone that the body uses at the cellular level. The bodies of some individuals are unable to do this conversion; consequently these people are given injections of T3. The half-life of T3 is about ten hours. Consider an individual who has taken 100 mcg of T3. a. How much of the T3 is in the body after thirty hours? b. Write an expression for the amount of T3 in the body after t hours. c. Use a graph to estimate the time required for the amount of T3 to be reduced to 10 mcg.

the hormone that the body uses at the cellular level. The bodies of some individuals are unable to do this conversion; consequently these people are given injections of T3. The half-life of T3 is about ten hours. Consider an individual who has taken 100 mcg of T3. a. How much of the T3 is in the body after thirty hours? b. Write an expression for the amount of T3 in the body after t hours. c. Use a graph to estimate the time required for the amount of T3 to be reduced to 10 mcg.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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year later if the money is compounded once a year, twice a year, and four times a year. timeters) of three fictional beers where t represents time (in seconds). b. Find an expression for the amount of money in the bank account if the money is compounded n times a year. Вeer 1: H (() 20(0.99)° Вeer 2: H(t) — 40 (0.9)' Вeer 3: H3 (1) 3D 15(0.98)° c. Evaluate this expression for larger and larger n and estimate the value it appears to be approach- ing. a. Which beer has the highest froth initially? What is the height? 33. Consider a bacterial species that produces ten off- spring per day. Assume that you start with one bac- terial cell, no cells die, and all cells reproduce at the b. Which beer has the slowest decay of froth? For this beer, what percentage of the height is lost in ten seconds? Twenty seconds? c. Which beer has the highest froth height after same rate. ten seconds? a. Compute the bacterial population size in one year if the cells reproduce only once per day, twice per day, and four times per day. 37. Hyperthyroidism is a condition in which the thyroid gland makes too much thyroid hormone. The condi- tion can lead to difficulty concentrating, fatigue, and weight loss. One treatment for hypothyroidism is the administration of replacement thyroid hormone such as thyroxine (T4). The concentration of this hormone in a patient's body exhibits exponential decay with a half-life of about seven days. Consider an individual that has taken 100 mcg of T4. b. Find an expression for the population size if the population reproduces n times per day. c. Evaluate this expression for larger and larger n and estimate the value it appears to be approach- ing. 34. Consider a bacterial species that produces five off- spring per day. Assume that you start with twenty bacterial cells, no cells die, and all cells reproduce at a. How much of the T4 is in the body after fourteen days? the same rate. a. Compute the bacterial population size in one year if the cells reproduce only once per day, twice per day, and four times per day. b. Write an expression for the amount of T4 in the body after t days. c. Use a graph to estimate the time required for the amount of T4 to be reduced to 10 mcg. b. Find an expression for the population size if the population reproduces n times per day. 38. In Problem 37, we discussed hyperthyroidism. For individuals with this conditions, the body converts
Transcribed Image Text:year later if the money is compounded once a year, twice a year, and four times a year. timeters) of three fictional beers where t represents time (in seconds). b. Find an expression for the amount of money in the bank account if the money is compounded n times a year. Вeer 1: H (() 20(0.99)° Вeer 2: H(t) — 40 (0.9)' Вeer 3: H3 (1) 3D 15(0.98)° c. Evaluate this expression for larger and larger n and estimate the value it appears to be approach- ing. a. Which beer has the highest froth initially? What is the height? 33. Consider a bacterial species that produces ten off- spring per day. Assume that you start with one bac- terial cell, no cells die, and all cells reproduce at the b. Which beer has the slowest decay of froth? For this beer, what percentage of the height is lost in ten seconds? Twenty seconds? c. Which beer has the highest froth height after same rate. ten seconds? a. Compute the bacterial population size in one year if the cells reproduce only once per day, twice per day, and four times per day. 37. Hyperthyroidism is a condition in which the thyroid gland makes too much thyroid hormone. The condi- tion can lead to difficulty concentrating, fatigue, and weight loss. One treatment for hypothyroidism is the administration of replacement thyroid hormone such as thyroxine (T4). The concentration of this hormone in a patient's body exhibits exponential decay with a half-life of about seven days. Consider an individual that has taken 100 mcg of T4. b. Find an expression for the population size if the population reproduces n times per day. c. Evaluate this expression for larger and larger n and estimate the value it appears to be approach- ing. 34. Consider a bacterial species that produces five off- spring per day. Assume that you start with twenty bacterial cells, no cells die, and all cells reproduce at a. How much of the T4 is in the body after fourteen days? the same rate. a. Compute the bacterial population size in one year if the cells reproduce only once per day, twice per day, and four times per day. b. Write an expression for the amount of T4 in the body after t days. c. Use a graph to estimate the time required for the amount of T4 to be reduced to 10 mcg. b. Find an expression for the population size if the population reproduces n times per day. 38. In Problem 37, we discussed hyperthyroidism. For individuals with this conditions, the body converts
1.4 Exponential Growth 57 43. Consider an instant lottery game in which you buy a scratch-off card and there is a certain chance p of winning a prize. If you buy N scratch-off cards, the chance of not winning a prize is (1 – p)N, that is, the chance you didn't win the prize on the first card times the chance you didn't win the prize on thyroxine (T4) into triiodothyronine (T3), which is the hormone that the body uses at the cellular level. The bodies of some individuals are unable to do this conversion; consequently these people are given injections of T3. The half-life of T3 is about - ten hours. Consider an individual who has taken 100 mcg of T3. the second card, and so on. Therefore, the chance of a. How much of the T3 is in the body after thirty winning a prize with N cards is 1 – (1 – p)N. a. If there is a 1/10 chance of winning a prize and you buy ten cards, what is the chance you win a prize? b. If there is a chance 1/100 of winning a prize and you buy 100 cards, what is the chance you win a prize? hours? b. Write an expression for the amount of T3 in the body after t hours. c. Use a graph to estimate the time required for the amount of T3 to be reduced to 10 mcg. 39. Carbon-14 has a half-life of 5,730 years. How much is left of 500 g of C-14 after t years? c. If there is chance 1/N of winning a prize and you buy N cards, what is the chance you win a prize? What value does this approach for large N? 40. If a bacterial population initially has twenty indi- viduals and doubles every 9.3 hours, then how many individuals will it have after three days? 41. "Whale Numbers up 12% a Year" was a headline in a 1993 Australian newspaper. In a thirteen-year study, beginning in 1981, scientists estimated that the humpback whale (Megaptera novaeangliae) population off the coast of Australia was increasing by 12% a year from a level of 350 individuals. 44. In an experimental study performed at Dartmouth College, two groups of mice with tumors were treated with the chemotherapeutic drug cisplatin. Prior to the therapy, the tumor consisted of prolif- erating cells (also known as clonogenic cells) that grew exponentially with a doubling time of approx- imately 2.9 days. Assume the initial tumor size was 0.1 cm³. a. Write an expression for P(t), the populations size at t years after 1981, assuming P(0) = 350. b. Estimate the size of the population in 2010. a. Write an expression for the tumor size after t days. b. Use graphing to estimate at what time the tumor is 0.5 cm³ in size. 42. The population size (in millions) of Mexico in the early 1980s is reported in Table 1.12. 45. In the experimental study described in Problem 44, each of the mice was given a dosage of 10 mg/kg of cisplatin. At the time of the therapy, the average tu- mor size was approximately 0.5 cm³. Assume all the Table 1 12 Population in Mexico
Transcribed Image Text:1.4 Exponential Growth 57 43. Consider an instant lottery game in which you buy a scratch-off card and there is a certain chance p of winning a prize. If you buy N scratch-off cards, the chance of not winning a prize is (1 – p)N, that is, the chance you didn't win the prize on the first card times the chance you didn't win the prize on thyroxine (T4) into triiodothyronine (T3), which is the hormone that the body uses at the cellular level. The bodies of some individuals are unable to do this conversion; consequently these people are given injections of T3. The half-life of T3 is about - ten hours. Consider an individual who has taken 100 mcg of T3. the second card, and so on. Therefore, the chance of a. How much of the T3 is in the body after thirty winning a prize with N cards is 1 – (1 – p)N. a. If there is a 1/10 chance of winning a prize and you buy ten cards, what is the chance you win a prize? b. If there is a chance 1/100 of winning a prize and you buy 100 cards, what is the chance you win a prize? hours? b. Write an expression for the amount of T3 in the body after t hours. c. Use a graph to estimate the time required for the amount of T3 to be reduced to 10 mcg. 39. Carbon-14 has a half-life of 5,730 years. How much is left of 500 g of C-14 after t years? c. If there is chance 1/N of winning a prize and you buy N cards, what is the chance you win a prize? What value does this approach for large N? 40. If a bacterial population initially has twenty indi- viduals and doubles every 9.3 hours, then how many individuals will it have after three days? 41. "Whale Numbers up 12% a Year" was a headline in a 1993 Australian newspaper. In a thirteen-year study, beginning in 1981, scientists estimated that the humpback whale (Megaptera novaeangliae) population off the coast of Australia was increasing by 12% a year from a level of 350 individuals. 44. In an experimental study performed at Dartmouth College, two groups of mice with tumors were treated with the chemotherapeutic drug cisplatin. Prior to the therapy, the tumor consisted of prolif- erating cells (also known as clonogenic cells) that grew exponentially with a doubling time of approx- imately 2.9 days. Assume the initial tumor size was 0.1 cm³. a. Write an expression for P(t), the populations size at t years after 1981, assuming P(0) = 350. b. Estimate the size of the population in 2010. a. Write an expression for the tumor size after t days. b. Use graphing to estimate at what time the tumor is 0.5 cm³ in size. 42. The population size (in millions) of Mexico in the early 1980s is reported in Table 1.12. 45. In the experimental study described in Problem 44, each of the mice was given a dosage of 10 mg/kg of cisplatin. At the time of the therapy, the average tu- mor size was approximately 0.5 cm³. Assume all the Table 1 12 Population in Mexico
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