Concept explainers
Blood Flow through the Brain One method of determining the level of blood flow through the brain requires the person to inhale air containing a fixed concentration of
nitrous oxide. During the first minute, the concentration of
after
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- Lidocaine Lidocaine is a drug used to treat irregular heartbeats. After an injection of a 100-mg dose of lidocaine, the amount of the drug in a patients bloodstream is 33.67(e0.0075731te0.12043t)mg at time t minutes after the injection. a. Make a graph of the amount of drug in the bloodstream for t up to 4 hours 240 minutes. b. When does the drug reach its maximum level in the bloodstream? c. For a person of typical size, the drug is effective as long as the amount in the bloodstream is at least 7.5 mg. For how long is the drug at or above that level? Hint: The drug is at that level twice. d. For a person of typical size, the lethal level occurs when the amount in the bloodstream exceeds 30 mg. Is this dose lethal for such a person? e. For a small person, the lethal level occurs when the amount in the bloodstream exceeds 15 mg. Is this dose lethal for such a person? If so, after how many minutes will it be lethal?arrow_forwardA Cup of Cofee The temperature C of a fresh cup of coffee t minutes after it is poured is given by C = 125e-0.03t 75 degree Fahrenheit. a. Make a graph of Cversus t. b. The coffee is cool enough to drink when its temperature if 150 degrees. When will the coffee be cool enough to drink? c. What is the temperature of the coffee in the pot? Note :We are assuming that the coffee pot is being kept hot and its the same temperature as the cup of coffee when it was poured. d.What is the temperature in the room where you are drinking the coffee? Hint: If the coffee is left to cool a long time, it will reach room temperature.arrow_forwardWorld Crude Oil Production In 1956, M.King Hubbert proposed a model to analyse crude oil production. His model, with updated data, gives world crude oil production as P=254.43e0.042t(1+2.12e0.042t)2 Here P is measured in billions of barrels per year, and t is time, in year, since 2000. a.Make a graph of world crude oil production for 2000 through 2040. b.When does this model predict a peak in world crude oil production? c.What is the maximum crude oil production predicted by this model?arrow_forward
- Buying a Car: If you buy a 25,000 car at an APR of 6, then your monthly payments depends on the term of the loan. If the term is t months, then your monthly payment is M=25,0000.0051.005t1.005t1dollars a. Make a graph of the monthly payments versus the length of the term, in months. Include terms from 12 to 60 months. b. Is the graph increasing or decreasing? What does that mean in practical terms? c. Do you lower your monthly payment more by changing the term of the loan from 12 to 24 months or from 48 to 60 months? Suggestion: you dont need a calculation here. The concavity of the graph can give you the answer.arrow_forwardBaking a Potato: A potato is placed in a preheated oven to bake. Its temperature P=P(t) is given by P=400325et/50, Where P is measured in degrees Fahrenheit and t is the time in minutes since the potato was placed in the oven. a. Make a graph of P versus t.Suggestion: in choosing your graphing window, it is reasonable to look at the potato over no more than a 2-hour period. After that, it will surely be burned to a crisp. You may wish to look at a table of values to select a vertical span. b. What was the initial temperature of the potato? c. Did the potatos temperature rise more during the first 30 minutes or second 30 minutes of baking? What was the average rate of change per minute during the first 30 minutes? What was the average rate of change per minute during the second 30 minutes. d. Is this graph concave up or concave down? Explain what that tells you about how the potato heats up, and relate this to part c.. e. The potato will be done when it reaches a temperature of 270 degrees. Approximate the time when the potato will be done. f. What is the temperature of the oven? Explain how you got your answer. Hint: if the potato were left in the oven for a long time, its temperature would match that of the oven.arrow_forwardCommute to Work Jade and her roommate Jari commute to work each morning, traveling west on I-10. One morning Jade left for work at 6:50 a.m., but Jari left 10 minutes later. Both drove at a constant speed. The following graphs show the distance in miles each of them has traveled on I-10 at time t in minutes, where t = 0 is 7 a.m. a Use the graph to decide which of them is traveling faster. b Find the speed in mi/h at which each of them is driving. c Find linear functions f and g that model the distances that Jade and Jari travel as functions of t in minutes.arrow_forward
- Monthly Payment for a Home: If you borrow 120,000 at an APR of 6 in order to buy a home, and if the lending institution compounds interest continuously, then your monthly payment M=M(Y), in dollars, depends on the number of years Y you take to pay off the loan. The relationship is given by M=120,000(e0.0051)1e0.06Y. a. Make a graph of M versus Y. in choosing a graphing window, you should note that a home mortgage rarely extends beyond 30 years. b. Express in functional notation your monthly payment if you pay off the loan in 20 years, and then use the graph to find that value. c. Use the graph to find your monthly payment if you pay off the loan in 30 years. d. From part b to part c of this exercise, you increased the debt period by 50. Did this decrease your monthly payment by 50? e. Is the graph concave up or concave down? Explain your answer in practical terms. f. Calculate the average decreases per year in your monthly payment from a loan period of 25 to a loan period of 30 years.arrow_forwardFluid Flow The intake pipe of a 100-gallon tank has a flow rate of 10 gallons per minute, and two drainpipes have flow rates of 5 gallons per minute each. The figure shows the volume V of fluid in the tank as a function of time t. Determine whether the input pipe and each drainpipe are open or closed in specific subintervals of the 1 hour of time shown in the graph. (There are many correct answers.)arrow_forwardTemperature An object at a temperature of 160C was removed from a furnace and placed in a room at 20C. The temperature T of the object was measured each hour h and recorded in the table. A model for the data is T=20+140e0.68h (a)The figure below shows the graph of the model. Use the graph to identify the horizontal asymptote of the model and interpret the asymptote in the context of the problem. (b) Use the model to approximate the time it took for the object to reach a temperature of 100C.arrow_forward
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