Concept explainers
Reminder Round all answers to two decimal places unless otherwise indicated.
A Train Each night at midnight a train leaves the station. The distance from a house to the train after t hours is given by
It is too noisy to sleep when the train is within 2 miles of the house. When is the train too near the house each night? Round your answers to the nearest minute. (Note: Be sure to convert your decimal answers to minutes.)
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Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
- Reminder Round all answers to two decimal places unless otherwise indicated. Note Some of the formulas below use the special number e, which was presented in the Prologue. A Roast is taken from the refrigerator where it had been for several days and placed immediately in a preheated oven to cook. The temperature R=R(t) of the roast t minutes after being placed in the oven is given by R=325280e0.005tdegreesFahrenheit a. What is the temperature of the refrigerator? b. Express the temperature of the roast 30 minutes after being put in the oven in functional notation, and then calculate its value. c. By how much did the temperature of the roast increase during the first 10 minutes of cooking? d. By how much did the temperature of the roast increase from the first hour to 10 minutes after the first hour of cooking?arrow_forwardReminderRound all answers to two decimal places unless otherwise indicated. NoteSome of the formulas below use the special number e, which was presented in the Prologue. Sound Pressure and DecibelsSounds exerts a pressure P on the human ear. This pressure increases as the loudness of the sound increases. If the loudness D is measure in decibels and the pressure P is measured in dynes per square centimeter, then the relationship is give by P=0.00021.122D. a.Ordinary conversation has a loudness of about 65 decibels. What is the pressure exerted on the human ear by ordinary conversation? b.A decibel level of 120 causes pain to the ear and can result in damage. What is the corresponding pressure level on the ear?arrow_forwardReminder: Round all answer to two decimal places unless otherwise indicated. 4.Thanksgiving Dinner It takes 10 minutes to preheat your oven to 325 degrees. For an oven preheated to 325 degrees, the recommended cooking time for a turkey is about 15 minutes per pound. a. Use a formula to express the total time T, in minutes, needed to preheat the oven and then bake a turkey weighing p pounds. b. Use your formula from part a to find the approximate time required to prepare a turkey weighing 18 pounds.arrow_forward
- Reminder Round all answers to decimal places unless otherwise indicated. Hiking You are hiking in a hilly region, and E=E(t) is your elevation at time t. a. Explain the meaning of dEdt in practical terms. b. Where might you be when dEdt is a large positive number? c. You reach a point where dEdt is briefly zero . Where might you be? d. Where might you be when dEdt is a large negative number?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. An Expensive Book A scientist visited Amazon.com to purchase a book. To his amazement, he found the book priced at 23,698,655.93. He discovered that there were two vendors selling this book on Amazon, and each was using a computer to adjust the price based on the other sellers price. The result was that each day the first vendors price was multiplied by 1.27. Assume that the initial price was 134.27, and determine how many days it took for the first vendors price to reach the exorbitant figure above. Round your answer to the nearest day.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Ship Propellers An ideal diameter d, in feet, of a ships propeller is given by the formula d=ch1/5r3/5. Here h is the horsepower of the engine driving the propeller. r is the maximum number of revolutions per minute of the propeller, and c is a constant. In both parts, give your answer in terms of a percentage. a. If the horsepower is increased by 20 while the number of revolutions per minute remains the same, how is the propeller diameter affected? b. If the horsepower remains the same while the number of revolutions per minute is increased by 20, how is the propeller diameter affected?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Newtons Law of Cooling says that a hot object cools rapidly when the difference between its temperature and that of the surrounding air is large, but the object cools more slowly when it nears room temperature. Suppose a piece of aluminum is removed from an oven and left to cool. The following table gives the temperature A=A(t), in degrees Fahrenheit. of the aluminum t minutes after it is removed from the own. a.Explain the meaning of A(75) and estimate its value. b.Find the average decrease of temperature pet minute during the first half-hour of cooling. t=Minutes A=Temperature 0 302 30 152 60 100 90 81 120 75 150 73 180 72 210 72 c.Find the average decrease per minute of temperature during the first half of the second hour of cooling. d.Explain how parts b and c support Newtons law of cooling. e.Use functional notation to express the temperature of the aluminum after I hour and 13 minutes. Estimate the temperature at that time. Note: Your work in part c should be helpful. f.What is the temperature of the oven? Exams your answer using functional notation, and give its value. g.Explain why you would expect the function A to have a limiting value. h.What is roan temperature? Explain your reasoning.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Catering a Dinner You are having a dinner catered. You pay a rental fee of 150 for the dining hall, and you pay the caterer 10 for each person who attends the dinner. a. Suppose you just want to break even. i.How much should you charge per ticket if you expect 50 people to attend? iiUse a formula to express the amount you should charge per ticket as a function of the number of people attending. Be sure to explain the meaning of the letters you choose and the units. iii.You expect 65 people to attend the dinner. Use your answer to part ii to express in functional notation the amount you should charge per ticket, and then calculate that amount. b. Suppose now that you want to make a profit of 100 from the dinner. Use a formula to express the amount you should charge per ticket as a function of the number of people attending. Again, be sure to explain the meaning of the letters you choose and the units.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Falling Objects An object is dropped near the surface of a planet. The time T, in seconds, it takes for the object to falls feet is given by the formula T=cs0.5 for some constant c that depends on the planet. a. On Earth, the constant c is 0.25. How long does it take an object to fall 16feet on Earth? b. On a certain planet, an object takes 5seconds to fall 20feet. What is the value of the constant c for this planet? c. If it takes 1second for an objects to fall 10feet on a planet, how long does it take an object to fall 40feet?arrow_forward
- Reminder Round all answers to two decimal places unless otherwise indicated. Cleaning Contaminated Water A tank of water is contaminated with 60 pounds of salt. In order to bring the salt concentration down to a level consistent with EPA standards, clean water is being piped into tank, and the well-mixed overflow is being collected for removal to a toxic-waste site. The result is that at the end of each hour, there is 22 less salt in the tank than at the beginning of the hour. Let S=S(t) denote the number of pounds of salt in the tank t hours after the flushing process begins. a. Explain why S is an exponential function and find its hourly decay factor. b. Give a formula for S. c. Make a graph of S that shows the flushing process during the first 15 hours, and describe in words how the salt removal process progresses. d. In order to meet EPA standards, there can be no more than 3 pounds of salt in the tank. How long must the process continue before EPA standards are met? e. Suppose this cleanup procedure costs 8000 per hour to operate. How much does it cost to reduce the amount of salt from 60 pounds to 3 pounds? How much does it cost to reduce the amount of salt from 3 pounds to 0.1 pound?arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Airspeed The airspeed of a plane is its speed in the absence of wind. With a headwind, ground speed the actual speed in relation to the ground is decreased by the speed of the wind. With a tailwind, ground speed is increased by the speed of the wind. Let A denote the airspeed of a plane and W the speed of the wind, both in miles per hour. Suppose it takes the plane 9 hours to travel the 720 miles from one town to another facing a headwind of W. the return trip, now with a tailwind of W, takes only 6 hours. a. Express the ground speed on the trip out in terms of A and W. b. Use the information from part a and the fact that distance equals rate times time to find an equation involving A and W for the trip out. c. Express the ground speed on the return trip in terms of A and W. d. Use the information from part c and the fact that distance equals rate times time to find an equation involving A and W for the return trip. e. Find the airspeed and the speed of the wind.arrow_forwardReminder Round all answers to two decimal places unless otherwise indicated. Sleeping Longer A certain man observed that each night he was sleeping 15 minutes longer than he had the night before, and he used this observation to predict the day of his death. 9 If he made his observation right after sleeping 8 hours, how long would it be until he slept 24 and so would never again wake?arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning